Properties of the set of continuous functions from acompact Hausdorff space to a topological field part II –
Example of an exotic topological field
Pawel [email protected]
Abstract Algebra Seminar, Knoxville, TN
December 12, 2016
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 1 / 1
Recall the previously discussed definitions and propositions:
Definition
Let X be a compact, Hausdorff space. Then C (X ,F ) denotes the set ofall continuous functions from X to F .
Definition
Let X be a compact, Hausdorff space and F a topological field. For eachy ∈ X , let
My := {f ∈ C (X ,F ) | f (y) = 0}.
Proposition
C (X ,F ) is a commutative ring with 1 6= 0 under the pointwise additionand multiplication of functions.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 2 / 1
Recall the previously discussed definitions and propositions:
Definition
Let X be a compact, Hausdorff space. Then C (X ,F ) denotes the set ofall continuous functions from X to F .
Definition
Let X be a compact, Hausdorff space and F a topological field. For eachy ∈ X , let
My := {f ∈ C (X ,F ) | f (y) = 0}.
Proposition
C (X ,F ) is a commutative ring with 1 6= 0 under the pointwise additionand multiplication of functions.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 2 / 1
Recall the previously discussed definitions and propositions:
Definition
Let X be a compact, Hausdorff space. Then C (X ,F ) denotes the set ofall continuous functions from X to F .
Definition
Let X be a compact, Hausdorff space and F a topological field. For eachy ∈ X , let
My := {f ∈ C (X ,F ) | f (y) = 0}.
Proposition
C (X ,F ) is a commutative ring with 1 6= 0 under the pointwise additionand multiplication of functions.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 2 / 1
Recall the previously discussed definitions and propositions:
Definition
Let X be a compact, Hausdorff space. Then C (X ,F ) denotes the set ofall continuous functions from X to F .
Definition
Let X be a compact, Hausdorff space and F a topological field. For eachy ∈ X , let
My := {f ∈ C (X ,F ) | f (y) = 0}.
Proposition
C (X ,F ) is a commutative ring with 1 6= 0 under the pointwise additionand multiplication of functions.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 2 / 1
Proposition
My is a maximal ideal of C (X ,F ) for all y ∈ X .
Proposition
Let R := C (X ,F ) and let
µ : X → Max (R)
be defined byµ(y) := My .
Regard Max (R) as a subspace of Spec (R) (endowed with Zariskitopology). Assume that {0} is a closed subset of F . Then µ is continuous.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 3 / 1
Proposition
My is a maximal ideal of C (X ,F ) for all y ∈ X .
Proposition
Let R := C (X ,F ) and let
µ : X → Max (R)
be defined byµ(y) := My .
Regard Max (R) as a subspace of Spec (R) (endowed with Zariskitopology). Assume that {0} is a closed subset of F . Then µ is continuous.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 3 / 1
Proposition
Assume that for each positive integer n, there is a polynomialPn ∈ F [X1, ...,Xn] such that
{(a1, ..., an) ∈ F n |Pn(a1, ..., an) = 0} = {(0, ..., 0)}.
Also, assume that {0} is a closed subset of F . Then the above map µ issurjective.
Proposition
Assume that given distinct x , y ∈ X , there is an f(x ,y) ∈ R such thatf(x ,y)(x) = 0 and f(x ,y)(y) 6= 0. Then the above map µ is injective.
Theorem
Assume F = R. Then the above map µ is a homeomorphism.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 4 / 1
Proposition
Assume that for each positive integer n, there is a polynomialPn ∈ F [X1, ...,Xn] such that
{(a1, ..., an) ∈ F n |Pn(a1, ..., an) = 0} = {(0, ..., 0)}.
Also, assume that {0} is a closed subset of F . Then the above map µ issurjective.
Proposition
Assume that given distinct x , y ∈ X , there is an f(x ,y) ∈ R such thatf(x ,y)(x) = 0 and f(x ,y)(y) 6= 0. Then the above map µ is injective.
Theorem
Assume F = R. Then the above map µ is a homeomorphism.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 4 / 1
Proposition
Assume that for each positive integer n, there is a polynomialPn ∈ F [X1, ...,Xn] such that
{(a1, ..., an) ∈ F n |Pn(a1, ..., an) = 0} = {(0, ..., 0)}.
Also, assume that {0} is a closed subset of F . Then the above map µ issurjective.
Proposition
Assume that given distinct x , y ∈ X , there is an f(x ,y) ∈ R such thatf(x ,y)(x) = 0 and f(x ,y)(y) 6= 0. Then the above map µ is injective.
Theorem
Assume F = R. Then the above map µ is a homeomorphism.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 4 / 1
Remark
The assumption of compactness of our Hausdorff space is crucial.
To seethat, let X = (0, 1) and let F = R. Then for each n ∈ Z, n ≥ 1, considerthe following function:
fn(x) =
1 x ≤ n
n+1
linear nn+1 ≤ x ≤ 2n+1
2n+2
0 x ≥ 2n+12n+2
(1)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 5 / 1
Remark
The assumption of compactness of our Hausdorff space is crucial. To seethat, let X = (0, 1) and let F = R.
Then for each n ∈ Z, n ≥ 1, considerthe following function:
fn(x) =
1 x ≤ n
n+1
linear nn+1 ≤ x ≤ 2n+1
2n+2
0 x ≥ 2n+12n+2
(1)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 5 / 1
Remark
The assumption of compactness of our Hausdorff space is crucial. To seethat, let X = (0, 1) and let F = R. Then for each n ∈ Z, n ≥ 1, considerthe following function:
fn(x) =
1 x ≤ n
n+1
linear nn+1 ≤ x ≤ 2n+1
2n+2
0 x ≥ 2n+12n+2
(1)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 5 / 1
Remark (Cont.)
Then consider the ideal generated by all such fn’s.
Call that ideal I . Thennotice that I 6= R, since if I = R, then 1 ∈ I , implying that
1 = g1fi1 + ...+ gmfim
for some g1, ..., gm ∈ R, fi1 , ..., fim ∈ I . But that cannot be, since if
N := max {i1, ..., im},
then for all x ≥ 2N+12N+2
fi1(x) = 0, ..., fim(x) = 0,
which in turn would imply that for any such x
1 = 1(x) = g1(x)fi1(x) + ...+ gm(x)fim(x) = 0,
a contradiction. Thus, I 6= R.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 6 / 1
Remark (Cont.)
Then consider the ideal generated by all such fn’s. Call that ideal I .
Thennotice that I 6= R, since if I = R, then 1 ∈ I , implying that
1 = g1fi1 + ...+ gmfim
for some g1, ..., gm ∈ R, fi1 , ..., fim ∈ I . But that cannot be, since if
N := max {i1, ..., im},
then for all x ≥ 2N+12N+2
fi1(x) = 0, ..., fim(x) = 0,
which in turn would imply that for any such x
1 = 1(x) = g1(x)fi1(x) + ...+ gm(x)fim(x) = 0,
a contradiction. Thus, I 6= R.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 6 / 1
Remark (Cont.)
Then consider the ideal generated by all such fn’s. Call that ideal I . Thennotice that I 6= R, since if I = R, then 1 ∈ I , implying that
1 = g1fi1 + ...+ gmfim
for some g1, ..., gm ∈ R, fi1 , ..., fim ∈ I .
But that cannot be, since if
N := max {i1, ..., im},
then for all x ≥ 2N+12N+2
fi1(x) = 0, ..., fim(x) = 0,
which in turn would imply that for any such x
1 = 1(x) = g1(x)fi1(x) + ...+ gm(x)fim(x) = 0,
a contradiction. Thus, I 6= R.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 6 / 1
Remark (Cont.)
Then consider the ideal generated by all such fn’s. Call that ideal I . Thennotice that I 6= R, since if I = R, then 1 ∈ I , implying that
1 = g1fi1 + ...+ gmfim
for some g1, ..., gm ∈ R, fi1 , ..., fim ∈ I . But that cannot be, since if
N := max {i1, ..., im},
then for all x ≥ 2N+12N+2
fi1(x) = 0, ..., fim(x) = 0,
which in turn would imply that for any such x
1 = 1(x) = g1(x)fi1(x) + ...+ gm(x)fim(x) = 0,
a contradiction. Thus, I 6= R.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 6 / 1
Remark (Cont.)
Then consider the ideal generated by all such fn’s. Call that ideal I . Thennotice that I 6= R, since if I = R, then 1 ∈ I , implying that
1 = g1fi1 + ...+ gmfim
for some g1, ..., gm ∈ R, fi1 , ..., fim ∈ I . But that cannot be, since if
N := max {i1, ..., im},
then for all x ≥ 2N+12N+2
fi1(x) = 0, ..., fim(x) = 0,
which in turn would imply that for any such x
1 = 1(x) = g1(x)fi1(x) + ...+ gm(x)fim(x) = 0,
a contradiction. Thus, I 6= R.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 6 / 1
Remark (Cont.)
Then consider the ideal generated by all such fn’s. Call that ideal I . Thennotice that I 6= R, since if I = R, then 1 ∈ I , implying that
1 = g1fi1 + ...+ gmfim
for some g1, ..., gm ∈ R, fi1 , ..., fim ∈ I . But that cannot be, since if
N := max {i1, ..., im},
then for all x ≥ 2N+12N+2
fi1(x) = 0, ..., fim(x) = 0,
which in turn would imply that for any such x
1 = 1(x) = g1(x)fi1(x) + ...+ gm(x)fim(x) = 0,
a contradiction.
Thus, I 6= R.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 6 / 1
Remark (Cont.)
Then consider the ideal generated by all such fn’s. Call that ideal I . Thennotice that I 6= R, since if I = R, then 1 ∈ I , implying that
1 = g1fi1 + ...+ gmfim
for some g1, ..., gm ∈ R, fi1 , ..., fim ∈ I . But that cannot be, since if
N := max {i1, ..., im},
then for all x ≥ 2N+12N+2
fi1(x) = 0, ..., fim(x) = 0,
which in turn would imply that for any such x
1 = 1(x) = g1(x)fi1(x) + ...+ gm(x)fim(x) = 0,
a contradiction. Thus, I 6= R.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 6 / 1
Remark (Cont.)
Since R is a nonzero ring with identity, I ⊂ M, where M is a maximalideal.
However, notice that M is not of the form My for any y ∈ X . For ifthat was the case, then all functions in M would vanish at y . But for anyy , we can find n big enough so that fn(y) 6= 0. Since all fn’s are in M, wereach a contradiction. Thus, the previously mentioned map µ is notsurjective.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 7 / 1
Remark (Cont.)
Since R is a nonzero ring with identity, I ⊂ M, where M is a maximalideal. However, notice that M is not of the form My for any y ∈ X .
For ifthat was the case, then all functions in M would vanish at y . But for anyy , we can find n big enough so that fn(y) 6= 0. Since all fn’s are in M, wereach a contradiction. Thus, the previously mentioned map µ is notsurjective.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 7 / 1
Remark (Cont.)
Since R is a nonzero ring with identity, I ⊂ M, where M is a maximalideal. However, notice that M is not of the form My for any y ∈ X . For ifthat was the case, then all functions in M would vanish at y . But for anyy , we can find n big enough so that fn(y) 6= 0.
Since all fn’s are in M, wereach a contradiction. Thus, the previously mentioned map µ is notsurjective.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 7 / 1
Remark (Cont.)
Since R is a nonzero ring with identity, I ⊂ M, where M is a maximalideal. However, notice that M is not of the form My for any y ∈ X . For ifthat was the case, then all functions in M would vanish at y . But for anyy , we can find n big enough so that fn(y) 6= 0. Since all fn’s are in M, wereach a contradiction.
Thus, the previously mentioned map µ is notsurjective.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 7 / 1
Remark (Cont.)
Since R is a nonzero ring with identity, I ⊂ M, where M is a maximalideal. However, notice that M is not of the form My for any y ∈ X . For ifthat was the case, then all functions in M would vanish at y . But for anyy , we can find n big enough so that fn(y) 6= 0. Since all fn’s are in M, wereach a contradiction. Thus, the previously mentioned map µ is notsurjective.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 7 / 1
Example
Let k be a field, let t be an indeterminate over k and let F := k((t)), i.e.the quotient field of the power series ring k[[t]].
It is a well-known factthat F coincides with the field of the formal Laurent series with coefficientfrom k defined by
k((t)) :=
∞∑
n≥Nant
n | an ∈ k , aN 6= 0, andN ∈ Z
∪ {0}Let ord denote the t-order, i.e.
ord
∞∑n≥N
antn
= N.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 8 / 1
Example
Let k be a field, let t be an indeterminate over k and let F := k((t)), i.e.the quotient field of the power series ring k[[t]]. It is a well-known factthat F coincides with the field of the formal Laurent series with coefficientfrom k defined by
k((t)) :=
∞∑
n≥Nant
n | an ∈ k , aN 6= 0, andN ∈ Z
∪ {0}
Let ord denote the t-order, i.e.
ord
∞∑n≥N
antn
= N.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 8 / 1
Example
Let k be a field, let t be an indeterminate over k and let F := k((t)), i.e.the quotient field of the power series ring k[[t]]. It is a well-known factthat F coincides with the field of the formal Laurent series with coefficientfrom k defined by
k((t)) :=
∞∑
n≥Nant
n | an ∈ k , aN 6= 0, andN ∈ Z
∪ {0}Let ord denote the t-order, i.e.
ord
∞∑n≥N
antn
= N.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 8 / 1
Remark
Notice that the order of a Laurent series is the minimum index for whichthe coefficient of that series is non-zero. Therefore, under the assumptionthat the minimum of the empty set is infinity, the zero polynomial hasorder infinity.
Proposition
Given f , g ∈ F , define the function d : F × F → [0,∞) by
d(f , g) :=1
2ord(f−g).
Then d is a metric on F .
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 9 / 1
Remark
Notice that the order of a Laurent series is the minimum index for whichthe coefficient of that series is non-zero. Therefore, under the assumptionthat the minimum of the empty set is infinity, the zero polynomial hasorder infinity.
Proposition
Given f , g ∈ F , define the function d : F × F → [0,∞) by
d(f , g) :=1
2ord(f−g).
Then d is a metric on F .
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 9 / 1
Proof.
First notice that the function f (x) = 12x is a non-negative function (it
attains zero at infinity). Therefore, d(f , g) ≥ 0 for all f , g ∈ F .
If f , g ∈ Fsuch that f = g , then ord(f − g) =∞, implying that d(f , g) = 0.Conversely, if d(f , g) = 0 for some f , g ∈ F , then ord(f − g) =∞,implying that f = g . To prove symmetry, notice that ord(f ) = ord(−f )for all f ∈ F , and thus
d(f , g) =1
2ord(f−g)=
1
2ord(−(f−g))=
1
2ord(g−f )= d(g , f )
Finally, we will present 2 proofs of the triangle inequality.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 10 / 1
Proof.
First notice that the function f (x) = 12x is a non-negative function (it
attains zero at infinity). Therefore, d(f , g) ≥ 0 for all f , g ∈ F . If f , g ∈ Fsuch that f = g , then ord(f − g) =∞, implying that d(f , g) = 0.
Conversely, if d(f , g) = 0 for some f , g ∈ F , then ord(f − g) =∞,implying that f = g . To prove symmetry, notice that ord(f ) = ord(−f )for all f ∈ F , and thus
d(f , g) =1
2ord(f−g)=
1
2ord(−(f−g))=
1
2ord(g−f )= d(g , f )
Finally, we will present 2 proofs of the triangle inequality.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 10 / 1
Proof.
First notice that the function f (x) = 12x is a non-negative function (it
attains zero at infinity). Therefore, d(f , g) ≥ 0 for all f , g ∈ F . If f , g ∈ Fsuch that f = g , then ord(f − g) =∞, implying that d(f , g) = 0.Conversely, if d(f , g) = 0 for some f , g ∈ F , then ord(f − g) =∞,implying that f = g .
To prove symmetry, notice that ord(f ) = ord(−f )for all f ∈ F , and thus
d(f , g) =1
2ord(f−g)=
1
2ord(−(f−g))=
1
2ord(g−f )= d(g , f )
Finally, we will present 2 proofs of the triangle inequality.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 10 / 1
Proof.
First notice that the function f (x) = 12x is a non-negative function (it
attains zero at infinity). Therefore, d(f , g) ≥ 0 for all f , g ∈ F . If f , g ∈ Fsuch that f = g , then ord(f − g) =∞, implying that d(f , g) = 0.Conversely, if d(f , g) = 0 for some f , g ∈ F , then ord(f − g) =∞,implying that f = g . To prove symmetry, notice that ord(f ) = ord(−f )for all f ∈ F , and thus
d(f , g) =1
2ord(f−g)=
1
2ord(−(f−g))=
1
2ord(g−f )= d(g , f )
Finally, we will present 2 proofs of the triangle inequality.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 10 / 1
Proof.
First notice that the function f (x) = 12x is a non-negative function (it
attains zero at infinity). Therefore, d(f , g) ≥ 0 for all f , g ∈ F . If f , g ∈ Fsuch that f = g , then ord(f − g) =∞, implying that d(f , g) = 0.Conversely, if d(f , g) = 0 for some f , g ∈ F , then ord(f − g) =∞,implying that f = g . To prove symmetry, notice that ord(f ) = ord(−f )for all f ∈ F , and thus
d(f , g) =1
2ord(f−g)=
1
2ord(−(f−g))=
1
2ord(g−f )= d(g , f )
Finally, we will present 2 proofs of the triangle inequality.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 10 / 1
Proof (Cont.)
Version I. Recall that for any a, b ∈ F , we have the following property:
ord(a + b) ≥ min {ord(a), ord(b).} (2)
Thus, it follows that for any f , g , h ∈ F , we have
ord(f − g) = ord(f − h + h − g)
≥ min{ord(f − h), ord(h − g)
}.
(3)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 11 / 1
Proof (Cont.)
Version I. Recall that for any a, b ∈ F , we have the following property:
ord(a + b) ≥ min {ord(a), ord(b).} (2)
Thus, it follows that for any f , g , h ∈ F , we have
ord(f − g) = ord(f − h + h − g)
≥ min{ord(f − h), ord(h − g)
}.
(3)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 11 / 1
Proof (Cont.)
Therefore,
d(f , g) =1
2ord(f−g)
≤ 1
2min{ord(f−h),ord(h−g)
}≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(4)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 12 / 1
Proof (Cont.)
Version II. Let f , g , h ∈ F be arbitrary. To show the triangle inequality,we will consider 2 cases:
Case I: ord(f ) < ord(g). This implies that ord(f − g) = ord(f ). Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h). In this case, ord(f − h) = ord(f ), and thus
d(f , g) =1
2ord(f−g)
=1
2ord(f )
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(5)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 13 / 1
Proof (Cont.)
Version II. Let f , g , h ∈ F be arbitrary. To show the triangle inequality,we will consider 2 cases:Case I: ord(f ) < ord(g).
This implies that ord(f − g) = ord(f ). Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h). In this case, ord(f − h) = ord(f ), and thus
d(f , g) =1
2ord(f−g)
=1
2ord(f )
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(5)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 13 / 1
Proof (Cont.)
Version II. Let f , g , h ∈ F be arbitrary. To show the triangle inequality,we will consider 2 cases:Case I: ord(f ) < ord(g). This implies that ord(f − g) = ord(f ).
Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h). In this case, ord(f − h) = ord(f ), and thus
d(f , g) =1
2ord(f−g)
=1
2ord(f )
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(5)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 13 / 1
Proof (Cont.)
Version II. Let f , g , h ∈ F be arbitrary. To show the triangle inequality,we will consider 2 cases:Case I: ord(f ) < ord(g). This implies that ord(f − g) = ord(f ). Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h).
In this case, ord(f − h) = ord(f ), and thus
d(f , g) =1
2ord(f−g)
=1
2ord(f )
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(5)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 13 / 1
Proof (Cont.)
Version II. Let f , g , h ∈ F be arbitrary. To show the triangle inequality,we will consider 2 cases:Case I: ord(f ) < ord(g). This implies that ord(f − g) = ord(f ). Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h). In this case, ord(f − h) = ord(f ),
and thus
d(f , g) =1
2ord(f−g)
=1
2ord(f )
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(5)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 13 / 1
Proof (Cont.)
Version II. Let f , g , h ∈ F be arbitrary. To show the triangle inequality,we will consider 2 cases:Case I: ord(f ) < ord(g). This implies that ord(f − g) = ord(f ). Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h). In this case, ord(f − h) = ord(f ), and thus
d(f , g) =1
2ord(f−g)
=1
2ord(f )
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(5)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 13 / 1
Proof (Cont.)
ord(f ) > ord(h).
In this case, ord(f − h) = ord(h), and thus
d(f , g) =1
2ord(f−g)
=1
2ord(f )
<1
2ord(h)
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(6)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 14 / 1
Proof (Cont.)
ord(f ) > ord(h). In this case, ord(f − h) = ord(h),
and thus
d(f , g) =1
2ord(f−g)
=1
2ord(f )
<1
2ord(h)
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(6)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 14 / 1
Proof (Cont.)
ord(f ) > ord(h). In this case, ord(f − h) = ord(h), and thus
d(f , g) =1
2ord(f−g)
=1
2ord(f )
<1
2ord(h)
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(6)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 14 / 1
Proof (Cont.)
ord(f ) = ord(h).
In this case, ord(f ) = ord(f − g) = ord(h− g), andthus
d(f , g) =1
2ord(f−g)
=1
2ord(h−g)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(7)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 15 / 1
Proof (Cont.)
ord(f ) = ord(h). In this case, ord(f ) = ord(f − g) = ord(h− g),
andthus
d(f , g) =1
2ord(f−g)
=1
2ord(h−g)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(7)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 15 / 1
Proof (Cont.)
ord(f ) = ord(h). In this case, ord(f ) = ord(f − g) = ord(h− g), andthus
d(f , g) =1
2ord(f−g)
=1
2ord(h−g)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(7)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 15 / 1
Proof (Cont.)
Case II: ord(f ) = ord(g).
This implies that ord(f − g) ≥ ord(f ). Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h). In this case, ord(f − h) = ord(f ), and thus
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f )
=1
2ord(f−h)
<1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(8)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 16 / 1
Proof (Cont.)
Case II: ord(f ) = ord(g). This implies that ord(f − g) ≥ ord(f ).
Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h). In this case, ord(f − h) = ord(f ), and thus
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f )
=1
2ord(f−h)
<1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(8)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 16 / 1
Proof (Cont.)
Case II: ord(f ) = ord(g). This implies that ord(f − g) ≥ ord(f ). Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h).
In this case, ord(f − h) = ord(f ), and thus
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f )
=1
2ord(f−h)
<1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(8)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 16 / 1
Proof (Cont.)
Case II: ord(f ) = ord(g). This implies that ord(f − g) ≥ ord(f ). Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h). In this case, ord(f − h) = ord(f ),
and thus
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f )
=1
2ord(f−h)
<1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(8)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 16 / 1
Proof (Cont.)
Case II: ord(f ) = ord(g). This implies that ord(f − g) ≥ ord(f ). Also, insuch a case, ord(h) can be one of the following:
ord(f ) < ord(h). In this case, ord(f − h) = ord(f ), and thus
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f )
=1
2ord(f−h)
<1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(8)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 16 / 1
Proof (Cont.)
ord(f ) > ord(h).
In this case, ord(f − h) = ord(h), and thus
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f )
<1
2ord(h)
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(9)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 17 / 1
Proof (Cont.)
ord(f ) > ord(h). In this case, ord(f − h) = ord(h),
and thus
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f )
<1
2ord(h)
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(9)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 17 / 1
Proof (Cont.)
ord(f ) > ord(h). In this case, ord(f − h) = ord(h), and thus
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f )
<1
2ord(h)
=1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(9)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 17 / 1
Proof (Cont.)
ord(f ) = ord(h).
In this case, let n := ord(f ) = ord(h) = ord(g) andlet k := ord(f − g). Then k ≥ n. That means that f and g share thefirst k − n coefficients. Now, if ord(f − h) ≤ ord(f − g) = k, then
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(10)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 18 / 1
Proof (Cont.)
ord(f ) = ord(h). In this case, let n := ord(f ) = ord(h) = ord(g) andlet k := ord(f − g).
Then k ≥ n. That means that f and g share thefirst k − n coefficients. Now, if ord(f − h) ≤ ord(f − g) = k, then
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(10)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 18 / 1
Proof (Cont.)
ord(f ) = ord(h). In this case, let n := ord(f ) = ord(h) = ord(g) andlet k := ord(f − g). Then k ≥ n.
That means that f and g share thefirst k − n coefficients. Now, if ord(f − h) ≤ ord(f − g) = k, then
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(10)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 18 / 1
Proof (Cont.)
ord(f ) = ord(h). In this case, let n := ord(f ) = ord(h) = ord(g) andlet k := ord(f − g). Then k ≥ n. That means that f and g share thefirst k − n coefficients.
Now, if ord(f − h) ≤ ord(f − g) = k, then
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(10)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 18 / 1
Proof (Cont.)
ord(f ) = ord(h). In this case, let n := ord(f ) = ord(h) = ord(g) andlet k := ord(f − g). Then k ≥ n. That means that f and g share thefirst k − n coefficients. Now, if ord(f − h) ≤ ord(f − g) = k, then
d(f , g) =1
2ord(f−g)
≤ 1
2ord(f−h)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
(10)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 18 / 1
Proof (Cont.)
If k ′ := ord(f − h) > ord(f − g) = k , then f and g share the first k ′ − ncoefficients.
In other words, f shares more first coefficients with h thanwith g . Since the first k ′ − n coefficents of f and h are the same, h sharesexactly as many first coefficients with g as f does. Thus,ord(f − g) = ord(h − g) and we can conclude that
d(f , g) =1
2ord(f−g)
=1
2ord(h−g)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
Remark
The case when ord(f ) > ord(g) follows from case I by symmetry.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 19 / 1
Proof (Cont.)
If k ′ := ord(f − h) > ord(f − g) = k , then f and g share the first k ′ − ncoefficients. In other words, f shares more first coefficients with h thanwith g .
Since the first k ′ − n coefficents of f and h are the same, h sharesexactly as many first coefficients with g as f does. Thus,ord(f − g) = ord(h − g) and we can conclude that
d(f , g) =1
2ord(f−g)
=1
2ord(h−g)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
Remark
The case when ord(f ) > ord(g) follows from case I by symmetry.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 19 / 1
Proof (Cont.)
If k ′ := ord(f − h) > ord(f − g) = k , then f and g share the first k ′ − ncoefficients. In other words, f shares more first coefficients with h thanwith g . Since the first k ′ − n coefficents of f and h are the same, h sharesexactly as many first coefficients with g as f does.
Thus,ord(f − g) = ord(h − g) and we can conclude that
d(f , g) =1
2ord(f−g)
=1
2ord(h−g)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
Remark
The case when ord(f ) > ord(g) follows from case I by symmetry.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 19 / 1
Proof (Cont.)
If k ′ := ord(f − h) > ord(f − g) = k , then f and g share the first k ′ − ncoefficients. In other words, f shares more first coefficients with h thanwith g . Since the first k ′ − n coefficents of f and h are the same, h sharesexactly as many first coefficients with g as f does. Thus,ord(f − g) = ord(h − g)
and we can conclude that
d(f , g) =1
2ord(f−g)
=1
2ord(h−g)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
Remark
The case when ord(f ) > ord(g) follows from case I by symmetry.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 19 / 1
Proof (Cont.)
If k ′ := ord(f − h) > ord(f − g) = k , then f and g share the first k ′ − ncoefficients. In other words, f shares more first coefficients with h thanwith g . Since the first k ′ − n coefficents of f and h are the same, h sharesexactly as many first coefficients with g as f does. Thus,ord(f − g) = ord(h − g) and we can conclude that
d(f , g) =1
2ord(f−g)
=1
2ord(h−g)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
Remark
The case when ord(f ) > ord(g) follows from case I by symmetry.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 19 / 1
Proof (Cont.)
If k ′ := ord(f − h) > ord(f − g) = k , then f and g share the first k ′ − ncoefficients. In other words, f shares more first coefficients with h thanwith g . Since the first k ′ − n coefficents of f and h are the same, h sharesexactly as many first coefficients with g as f does. Thus,ord(f − g) = ord(h − g) and we can conclude that
d(f , g) =1
2ord(f−g)
=1
2ord(h−g)
≤ 1
2ord(f−h)+
1
2ord(h−g)
= d(f , h) + d(h, g).
Remark
The case when ord(f ) > ord(g) follows from case I by symmetry.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 19 / 1
Proposition
Under the metric topology resulting from the above metric, F is atopological field.
Proof.
To show that F is a topological field, we need to show that addition,multiplication, and reciprocal maps are continuous.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 20 / 1
Proposition
Under the metric topology resulting from the above metric, F is atopological field.
Proof.
To show that F is a topological field, we need to show that addition,multiplication, and reciprocal maps are continuous.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 20 / 1
Proof (Cont.)
Case I: To show that addition is continuous, let (fn, gn)n→∞−−−→ (f , g) in
the product topology induced by the above metric.
Recall that thishappens if and only if
fnn→∞−−−→ f and gn
n→∞−−−→ g .
Notice that this is equivalent to saying that
d(fn, f )n→∞−−−→ 0 and d(gn, g)
n→∞−−−→ 0.
But from the definition of our metric, this is equivalent to saying that
ord(fn − f )n→∞−−−→∞ and ord(gn − g)
n→∞−−−→∞,
i.e. fn (respectively gn) has more and more ”first” coefficients exactlysame as f (respectively g) as n goes to infinity.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 21 / 1
Proof (Cont.)
Case I: To show that addition is continuous, let (fn, gn)n→∞−−−→ (f , g) in
the product topology induced by the above metric. Recall that thishappens if and only if
fnn→∞−−−→ f and gn
n→∞−−−→ g .
Notice that this is equivalent to saying that
d(fn, f )n→∞−−−→ 0 and d(gn, g)
n→∞−−−→ 0.
But from the definition of our metric, this is equivalent to saying that
ord(fn − f )n→∞−−−→∞ and ord(gn − g)
n→∞−−−→∞,
i.e. fn (respectively gn) has more and more ”first” coefficients exactlysame as f (respectively g) as n goes to infinity.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 21 / 1
Proof (Cont.)
Case I: To show that addition is continuous, let (fn, gn)n→∞−−−→ (f , g) in
the product topology induced by the above metric. Recall that thishappens if and only if
fnn→∞−−−→ f and gn
n→∞−−−→ g .
Notice that this is equivalent to saying that
d(fn, f )n→∞−−−→ 0 and d(gn, g)
n→∞−−−→ 0.
But from the definition of our metric, this is equivalent to saying that
ord(fn − f )n→∞−−−→∞ and ord(gn − g)
n→∞−−−→∞,
i.e. fn (respectively gn) has more and more ”first” coefficients exactlysame as f (respectively g) as n goes to infinity.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 21 / 1
Proof (Cont.)
Case I: To show that addition is continuous, let (fn, gn)n→∞−−−→ (f , g) in
the product topology induced by the above metric. Recall that thishappens if and only if
fnn→∞−−−→ f and gn
n→∞−−−→ g .
Notice that this is equivalent to saying that
d(fn, f )n→∞−−−→ 0 and d(gn, g)
n→∞−−−→ 0.
But from the definition of our metric, this is equivalent to saying that
ord(fn − f )n→∞−−−→∞ and ord(gn − g)
n→∞−−−→∞,
i.e. fn (respectively gn) has more and more ”first” coefficients exactlysame as f (respectively g) as n goes to infinity.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 21 / 1
Proof (Cont.)
Case I: To show that addition is continuous, let (fn, gn)n→∞−−−→ (f , g) in
the product topology induced by the above metric. Recall that thishappens if and only if
fnn→∞−−−→ f and gn
n→∞−−−→ g .
Notice that this is equivalent to saying that
d(fn, f )n→∞−−−→ 0 and d(gn, g)
n→∞−−−→ 0.
But from the definition of our metric, this is equivalent to saying that
ord(fn − f )n→∞−−−→∞ and ord(gn − g)
n→∞−−−→∞,
i.e. fn (respectively gn) has more and more ”first” coefficients exactlysame as f (respectively g) as n goes to infinity.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 21 / 1
Proof (Cont.)
Thus, from the above discussion, to show that
fn + gnn→∞−−−→ f + g ,
it is enough to show that
ord(fn + gn − (f + g))n→∞−−−→∞.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 22 / 1
Proof (Cont.)
To do so, recall that for any a, b ∈ F , we have the following property:
ord(a + b) ≥ min {ord(a), ord(b)} (11)
Therefore, we see that
ord(fn + gn − (f + g)) = ord(fn − f + g − gn)
≥ min {ord(fn − f ), ord(g − gn)}= min {ord(fn − f ), ord(gn − g)},
(12)
and since both orders on the right go to infinity as n goes to infinity, sodoes the minimum, and thus as n goes to infinity, the order offn + gn − (f + g) goes to infinity as well.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 23 / 1
Proof (Cont.)
To do so, recall that for any a, b ∈ F , we have the following property:
ord(a + b) ≥ min {ord(a), ord(b)} (11)
Therefore, we see that
ord(fn + gn − (f + g)) = ord(fn − f + g − gn)
≥ min {ord(fn − f ), ord(g − gn)}= min {ord(fn − f ), ord(gn − g)},
(12)
and since both orders on the right go to infinity as n goes to infinity, sodoes the minimum, and thus as n goes to infinity, the order offn + gn − (f + g) goes to infinity as well.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 23 / 1
Proof (Cont.)
To do so, recall that for any a, b ∈ F , we have the following property:
ord(a + b) ≥ min {ord(a), ord(b)} (11)
Therefore, we see that
ord(fn + gn − (f + g)) = ord(fn − f + g − gn)
≥ min {ord(fn − f ), ord(g − gn)}= min {ord(fn − f ), ord(gn − g)},
(12)
and since both orders on the right go to infinity as n goes to infinity, sodoes the minimum, and thus as n goes to infinity, the order offn + gn − (f + g) goes to infinity as well.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 23 / 1
Proof (Cont.)
Case II: To show that multiplication is continuous, let fn and gn be asabove.
By the above discussion, it is enough to show that
ord(fngn − fg)n→∞−−−→∞
To do so, notice that if gnn→∞−−−→ g , then ord(gn − g)
n→∞−−−→∞, which inturn implies that if g 6= 0, then for a sufficiently large n,
ord(gn) = ord(g),
and if g = 0, thenord(gn)
n→∞−−−→∞.
Also, recall that since F is a field (and thus an integral domain), it has nozero divisors, and therefore for any a, b ∈ F ,
ord(ab) = ord(a) + ord(b) (13)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 24 / 1
Proof (Cont.)
Case II: To show that multiplication is continuous, let fn and gn be asabove. By the above discussion, it is enough to show that
ord(fngn − fg)n→∞−−−→∞
To do so, notice that if gnn→∞−−−→ g , then ord(gn − g)
n→∞−−−→∞, which inturn implies that if g 6= 0, then for a sufficiently large n,
ord(gn) = ord(g),
and if g = 0, thenord(gn)
n→∞−−−→∞.
Also, recall that since F is a field (and thus an integral domain), it has nozero divisors, and therefore for any a, b ∈ F ,
ord(ab) = ord(a) + ord(b) (13)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 24 / 1
Proof (Cont.)
Case II: To show that multiplication is continuous, let fn and gn be asabove. By the above discussion, it is enough to show that
ord(fngn − fg)n→∞−−−→∞
To do so, notice that if gnn→∞−−−→ g , then ord(gn − g)
n→∞−−−→∞,
which inturn implies that if g 6= 0, then for a sufficiently large n,
ord(gn) = ord(g),
and if g = 0, thenord(gn)
n→∞−−−→∞.
Also, recall that since F is a field (and thus an integral domain), it has nozero divisors, and therefore for any a, b ∈ F ,
ord(ab) = ord(a) + ord(b) (13)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 24 / 1
Proof (Cont.)
Case II: To show that multiplication is continuous, let fn and gn be asabove. By the above discussion, it is enough to show that
ord(fngn − fg)n→∞−−−→∞
To do so, notice that if gnn→∞−−−→ g , then ord(gn − g)
n→∞−−−→∞, which inturn implies that if g 6= 0, then for a sufficiently large n,
ord(gn) = ord(g),
and if g = 0, thenord(gn)
n→∞−−−→∞.
Also, recall that since F is a field (and thus an integral domain), it has nozero divisors, and therefore for any a, b ∈ F ,
ord(ab) = ord(a) + ord(b) (13)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 24 / 1
Proof (Cont.)
Case II: To show that multiplication is continuous, let fn and gn be asabove. By the above discussion, it is enough to show that
ord(fngn − fg)n→∞−−−→∞
To do so, notice that if gnn→∞−−−→ g , then ord(gn − g)
n→∞−−−→∞, which inturn implies that if g 6= 0, then for a sufficiently large n,
ord(gn) = ord(g),
and if g = 0, thenord(gn)
n→∞−−−→∞.
Also, recall that since F is a field (and thus an integral domain), it has nozero divisors, and therefore for any a, b ∈ F ,
ord(ab) = ord(a) + ord(b) (13)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 24 / 1
Proof (Cont.)
Case II: To show that multiplication is continuous, let fn and gn be asabove. By the above discussion, it is enough to show that
ord(fngn − fg)n→∞−−−→∞
To do so, notice that if gnn→∞−−−→ g , then ord(gn − g)
n→∞−−−→∞, which inturn implies that if g 6= 0, then for a sufficiently large n,
ord(gn) = ord(g),
and if g = 0, thenord(gn)
n→∞−−−→∞.
Also, recall that since F is a field (and thus an integral domain), it has nozero divisors, and therefore for any a, b ∈ F ,
ord(ab) = ord(a) + ord(b) (13)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 24 / 1
Proof (Cont.)
These two facts help us notice that
ord(fngn − fg) = ord(fngn − fgn + fgn − fg)
= ord((fn − f )gn + f (gn − g))
≥ min {ord((fn − f )gn), ord(f (gn − g))},≥ min {ord(fn − f ) + ord(gn), ord(f ) + ord(gn − g)}.
(14)
But as n goes to infinity, ord(gn) becomes fixed or goes to infinity, ord(f )is fixed, and both ord(fn − f ) and ord(gn − g) go to infinity. Therefore,this implies that
ord(fngn − fg)n→∞−−−→∞.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 25 / 1
Proof (Cont.)
These two facts help us notice that
ord(fngn − fg) = ord(fngn − fgn + fgn − fg)
= ord((fn − f )gn + f (gn − g))
≥ min {ord((fn − f )gn), ord(f (gn − g))},≥ min {ord(fn − f ) + ord(gn), ord(f ) + ord(gn − g)}.
(14)
But as n goes to infinity, ord(gn) becomes fixed or goes to infinity, ord(f )is fixed, and both ord(fn − f ) and ord(gn − g) go to infinity. Therefore,this implies that
ord(fngn − fg)n→∞−−−→∞.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 25 / 1
Proof (Cont.)
Case III: To show that the reciprocal map is continuous, let fnn→∞−−−→ f ,
where f 6= 0 and fn 6= 0 for all n.
By the above discussion, it is enough toshow that
ord
(1
fn− 1
f
)n→∞−−−→∞
Recall that since F is a field (and thus an integral domain), it has no zerodivisors, and therefore for any a, b ∈ F such that b 6= 0, we have
ord(ab
)= ord(a)− ord(b) (15)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 26 / 1
Proof (Cont.)
Case III: To show that the reciprocal map is continuous, let fnn→∞−−−→ f ,
where f 6= 0 and fn 6= 0 for all n. By the above discussion, it is enough toshow that
ord
(1
fn− 1
f
)n→∞−−−→∞
Recall that since F is a field (and thus an integral domain), it has no zerodivisors, and therefore for any a, b ∈ F such that b 6= 0, we have
ord(ab
)= ord(a)− ord(b) (15)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 26 / 1
Proof (Cont.)
Case III: To show that the reciprocal map is continuous, let fnn→∞−−−→ f ,
where f 6= 0 and fn 6= 0 for all n. By the above discussion, it is enough toshow that
ord
(1
fn− 1
f
)n→∞−−−→∞
Recall that since F is a field (and thus an integral domain), it has no zerodivisors, and therefore for any a, b ∈ F such that b 6= 0, we have
ord(ab
)= ord(a)− ord(b) (15)
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 26 / 1
Proof (Cont.)
Therefore, by using this fact and facts established before, we see that
ord
(1
fn− 1
f
)= ord
(f − fnfnf
)= ord(fn − f )− ord(fnf )
= ord(fn − f )− (ord(fn) + ord(f ))
(16)
Since f 6= 0, order of fn becomes fixed for big enough n and ord(fn − f )goes to infinity, the left side also approaches infinity an n increases. Thisfinishes the proof that F is a topological field.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 27 / 1
Proof (Cont.)
Therefore, by using this fact and facts established before, we see that
ord
(1
fn− 1
f
)= ord
(f − fnfnf
)= ord(fn − f )− ord(fnf )
= ord(fn − f )− (ord(fn) + ord(f ))
(16)
Since f 6= 0, order of fn becomes fixed for big enough n and ord(fn − f )goes to infinity, the left side also approaches infinity an n increases. Thisfinishes the proof that F is a topological field.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 27 / 1
Proposition
Let P2 := X 22 − tX 2
1 and inductively define
Pn := X 2n − tP2
n−1 for all integers n ≥ 3
Then for (a1, ..., an) ∈ F n, we have
Pn(a1, ..., an) = 0 iff (a1, ..., an) = (0, ..., 0).
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 28 / 1
Proof.
First, let (a1, ..., an) = (0, ..., 0). Then we proceed by induction:
Base case: If k = 2, then
P2(0, 0) = 02 − t · 02 = 0.
Inductive step: Assume that the statement holds for k − 1 < n. Then
Pk(0, ..., 0) = 02 − t(Pk−1(0, ..., 0)
)2= 0.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 29 / 1
Proof.
First, let (a1, ..., an) = (0, ..., 0). Then we proceed by induction:Base case: If k = 2, then
P2(0, 0) = 02 − t · 02 = 0.
Inductive step: Assume that the statement holds for k − 1 < n. Then
Pk(0, ..., 0) = 02 − t(Pk−1(0, ..., 0)
)2= 0.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 29 / 1
Proof.
First, let (a1, ..., an) = (0, ..., 0). Then we proceed by induction:Base case: If k = 2, then
P2(0, 0) = 02 − t · 02 = 0.
Inductive step: Assume that the statement holds for k − 1 < n. Then
Pk(0, ..., 0) = 02 − t(Pk−1(0, ..., 0)
)2= 0.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 29 / 1
Proof (Cont.)
Conversely, let (a1, ..., an) be such that Pn(a1, ..., an) = 0.
Then
Pn(a1, ..., an) = a2n − t
(Pn−1(a1, ..., an−1)
)2= 0,
which is equivalent to
a2n = t
(Pn−1(a1, ..., an−1)
)2.
However, if the order of an is finite, we see that the t-order on the left sideis even, whereas the t-order on the right side is odd. Thus, order has to beinfinite, implying that an = 0 and Pn−1(a1, ..., an−1) = 0.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 30 / 1
Proof (Cont.)
Conversely, let (a1, ..., an) be such that Pn(a1, ..., an) = 0. Then
Pn(a1, ..., an) = a2n − t
(Pn−1(a1, ..., an−1)
)2= 0,
which is equivalent to
a2n = t
(Pn−1(a1, ..., an−1)
)2.
However, if the order of an is finite, we see that the t-order on the left sideis even, whereas the t-order on the right side is odd. Thus, order has to beinfinite, implying that an = 0 and Pn−1(a1, ..., an−1) = 0.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 30 / 1
Proof (Cont.)
Conversely, let (a1, ..., an) be such that Pn(a1, ..., an) = 0. Then
Pn(a1, ..., an) = a2n − t
(Pn−1(a1, ..., an−1)
)2= 0,
which is equivalent to
a2n = t
(Pn−1(a1, ..., an−1)
)2.
However, if the order of an is finite, we see that the t-order on the left sideis even, whereas the t-order on the right side is odd. Thus, order has to beinfinite, implying that an = 0 and Pn−1(a1, ..., an−1) = 0.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 30 / 1
Proof (Cont.)
Conversely, let (a1, ..., an) be such that Pn(a1, ..., an) = 0. Then
Pn(a1, ..., an) = a2n − t
(Pn−1(a1, ..., an−1)
)2= 0,
which is equivalent to
a2n = t
(Pn−1(a1, ..., an−1)
)2.
However, if the order of an is finite, we see that the t-order on the left sideis even, whereas the t-order on the right side is odd.
Thus, order has to beinfinite, implying that an = 0 and Pn−1(a1, ..., an−1) = 0.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 30 / 1
Proof (Cont.)
Conversely, let (a1, ..., an) be such that Pn(a1, ..., an) = 0. Then
Pn(a1, ..., an) = a2n − t
(Pn−1(a1, ..., an−1)
)2= 0,
which is equivalent to
a2n = t
(Pn−1(a1, ..., an−1)
)2.
However, if the order of an is finite, we see that the t-order on the left sideis even, whereas the t-order on the right side is odd. Thus, order has to beinfinite, implying that an = 0 and Pn−1(a1, ..., an−1) = 0.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 30 / 1
Proof (Cont.)
Continuing in this manner, we get that
(an, ..., a3) = 0 and P2(a1, a2) = a22 − ta2
1 = 0
But by applying the same logic, we see that (a1, a2) = (0, 0), implying that
(a1, ..., an) = (0, ..., 0),
as desired.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 31 / 1
Proof (Cont.)
Continuing in this manner, we get that
(an, ..., a3) = 0 and P2(a1, a2) = a22 − ta2
1 = 0
But by applying the same logic, we see that (a1, a2) = (0, 0), implying that
(a1, ..., an) = (0, ..., 0),
as desired.
Pawel Grzegrzolka [email protected] Properties of the set of continuous functions from a compact Hausdorff space to a topological field part II – Example of an exotic topological fieldDecember 12, 2016 31 / 1