counterexamples in topology

8/10/2019 Counterexamples in Topology

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Counterexamples in Topology

Counterexamples in Topology (1970, 2nd ed. 1978) isa book on mathematics by topologists Lynn Steen and J.Arthur Seebach, Jr.In the process of working on problems like themetrization problem, topologists (including Steen andSeebach) have dened a wide variety of topological prop-erties. It is often useful in the study and understanding ofabstracts such as topological spaces to determine that oneproperty does not follow from another. One of the easi-est ways of doing this is to nd a counterexample whichexhibits one property but not the other. In Counterexam- ples in Topology , Steen and Seebach, together with vestudents in an undergraduate research project at St. OlafCollege, Minnesotain thesummer of 1967, canvassed theeld of topology for such counterexamples and compiledthem in an attempt to simplify the literature.For instance, an example of a rst-countable spacewhich is not second-countable is counterexample #3, thediscrete topology on an uncountable set. This particularcounterexample shows that second-countability does notfollow from rst-countability.

Several other Counterexamples in ... books and papershave followed, with similar motivations.

1 Notation

Several of the naming conventions in this book differfrom more accepted modern conventions, particularlywith respect to the separation axioms. The authors usethe terms T3 , T4 , and T5 to refer to regular, normal, andcompletely normal. They also refer to completely Haus-dorffasUrysohn. This was a result of the different histor-

ical development of metrization theory andgeneral topol-ogy; see History of the separation axioms for more.

2 List of mentioned counterexam-ples

1. Finite discrete topology

2. Countable discrete topology

3. Uncountable discrete topology

4. Indiscrete topology

5. Partition topology

6. Oddeven topology

7. Deleted integer topology

8. Finite particular point topology

9. Countable particular point topology

10. Uncountable particular point topology

11. Sierpinski space, see also particular point topology

12. Closed extension topology13. Finite excluded point topology

14. Countable excluded point topology

15. Uncountable excluded point topology

16. Open extension topology

17. Either-or topology

18. Finite complement topology on a countable space

19. Finite complement topology on an uncountablespace

20. Countable complement topology

21. Double pointed countable complement topology

22. Compact complement topology

23. Countable Fort space

24. Uncountable Fort space

25. Fortissimo space

26. ArensFort space

27. Modied Fort space

28. Euclidean topology

29. Cantor set

30. Rational numbers

31. Irrational numbers

32. Special subsets of the real line

33. Special subsets of the plane

34. One point compactication topology

35. One point compactication of the rationals

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https://en.wikipedia.org/wiki/One_point_compactificationhttps://en.wikipedia.org/wiki/Irrational_numberhttps://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Cantor_sethttps://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Fort_spacehttps://en.wikipedia.org/wiki/Arens%E2%80%93Fort_spacehttps://en.wikipedia.org/wiki/Fortissimo_spacehttps://en.wikipedia.org/wiki/Fort_spacehttps://en.wikipedia.org/wiki/Fort_spacehttps://en.wikipedia.org/wiki/Compact_complement_topologyhttps://en.wikipedia.org/wiki/Countable_complement_topologyhttps://en.wikipedia.org/wiki/Countable_complement_topologyhttps://en.wikipedia.org/wiki/Finite_complement_topologyhttps://en.wikipedia.org/wiki/Countablehttps://en.wikipedia.org/wiki/Finite_complement_topologyhttps://en.wikipedia.org/wiki/Either-or_topologyhttps://en.wikipedia.org/wiki/Open_extension_topologyhttps://en.wikipedia.org/wiki/Excluded_point_topologyhttps://en.wikipedia.org/wiki/Excluded_point_topologyhttps://en.wikipedia.org/wiki/Excluded_point_topologyhttps://en.wikipedia.org/wiki/Closed_extension_topologyhttps://en.wikipedia.org/wiki/Particular_point_topologyhttps://en.wikipedia.org/wiki/Sierpinski_spacehttps://en.wikipedia.org/wiki/Particular_point_topologyhttps://en.wikipedia.org/wiki/Particular_point_topologyhttps://en.wikipedia.org/wiki/Particular_point_topologyhttps://en.wikipedia.org/wiki/Deleted_integer_topologyhttps://en.wikipedia.org/wiki/Odd%E2%80%93even_topologyhttps://en.wikipedia.org/wiki/Partition_topologyhttps://en.wikipedia.org/wiki/Indiscrete_topologyhttps://en.wikipedia.org/wiki/Discrete_topologyhttps://en.wikipedia.org/wiki/Discrete_topologyhttps://en.wikipedia.org/wiki/Countablehttps://en.wikipedia.org/wiki/Discrete_topologyhttps://en.wikipedia.org/wiki/Finite_sethttps://en.wikipedia.org/wiki/History_of_the_separation_axiomshttps://en.wikipedia.org/wiki/General_topologyhttps://en.wikipedia.org/wiki/General_topologyhttps://en.wikipedia.org/wiki/Urysohn_spacehttps://en.wikipedia.org/wiki/Completely_Hausdorff_spacehttps://en.wikipedia.org/wiki/Completely_Hausdorff_spacehttps://en.wikipedia.org/wiki/Completely_normal_spacehttps://en.wikipedia.org/wiki/Normal_spacehttps://en.wikipedia.org/wiki/Regular_spacehttps://en.wikipedia.org/wiki/Separation_axiomhttps://en.wikipedia.org/wiki/Naming_conventionhttps://en.wikipedia.org/wiki/Uncountable_sethttps://en.wikipedia.org/wiki/Discrete_topologyhttps://en.wikipedia.org/wiki/Second-countable_spacehttps://en.wikipedia.org/wiki/First-countable_spacehttps://en.wikipedia.org/wiki/Topologyhttps://en.wikipedia.org/wiki/Minnesotahttps://en.wikipedia.org/wiki/St._Olaf_Collegehttps://en.wikipedia.org/wiki/St._Olaf_Collegehttps://en.wikipedia.org/wiki/Counterexamplehttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Topological_propertieshttps://en.wikipedia.org/wiki/Topological_propertieshttps://en.wikipedia.org/wiki/Metrization_problemhttps://en.wikipedia.org/wiki/J._Arthur_Seebach,_Jr.https://en.wikipedia.org/wiki/J._Arthur_Seebach,_Jr.https://en.wikipedia.org/wiki/Lynn_Steenhttps://en.wikipedia.org/wiki/Topologyhttps://en.wikipedia.org/wiki/Mathematics


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108. C[0,1]

109. Box product topology on R

110. Stoneech compactication

111. Stoneech compactication of the integers

112. Novak space

113. Strong ultralter topology

114. Single ultralter topology

115. Nested rectangles

116. Topologists sine curve

117. Closed topologists sine curve

118. Extended topologists sine curve

119. Innite broom120. Closed innite broom

121. Integer broom

122. Nested angles

123. Innite cage

124. Bernsteins connected sets

125. Gustins sequence space

126. Roys lattice space

127. Roys lattice subspace

128. Cantors leaky tent

129. Cantors teepee

130. Pseudo-arc

131. Millers biconnected set

132. Wheel without its hub

133. Tangoras connected space

134. Bounded metrics135. Sierpinskis metric space

136. Duncans space

137. Cauchy completion

138. Hausdorffs metric topology

139. Post Office metric

140. Radial metric

141. Radial interval topology

142. Bings discrete extension space

143. Michaels closed subspace

3 See also

List of examples in general topology

-Base: An Interactive Encyclopediaof TopologicalSpaces

4 References

Lynn ArthurSteenandJ. ArthurSeebach, Jr.,Coun-terexamples in Topology . Springer-Verlag, NewYork, 1978. Reprinted by Dover Publications, NewYork, 1995. ISBN 0-486-68735-X (Dover edition).
https://en.wikipedia.org/wiki/Special:BookSources/048668735Xhttp://topology.jdabbs.com/http://topology.jdabbs.com/https://en.wikipedia.org/wiki/List_of_examples_in_general_topologyhttps://en.wikipedia.org/wiki/Michael%2527s_closed_subspacehttps://en.wikipedia.org/wiki/Bing%2527s_discrete_extension_spacehttps://en.wikipedia.org/wiki/Post_Office_metrichttps://en.wikipedia.org/wiki/Hausdorff_distancehttps://en.wikipedia.org/wiki/Cauchy_completionhttps://en.wikipedia.org/wiki/Duncan%2527s_spacehttps://en.wikipedia.org/wiki/Sierpinski%2527s_metric_spacehttps://en.wikipedia.org/wiki/Tangora%2527s_connected_spacehttps://en.wikipedia.org/wiki/Miller%2527s_biconnected_sethttps://en.wikipedia.org/wiki/Pseudo-archttps://en.wikipedia.org/wiki/Cantor%2527s_teepeehttps://en.wikipedia.org/wiki/Knaster-Kuratowski_fanhttps://en.wikipedia.org/wiki/Roy%2527s_lattice_subspacehttps://en.wikipedia.org/wiki/Roy%2527s_lattice_spacehttps://en.wikipedia.org/wiki/Gustin%2527s_sequence_spacehttps://en.wikipedia.org/wiki/Bernstein%2527s_connected_setshttps://en.wikipedia.org/wiki/Integer_broomhttps://en.wikipedia.org/wiki/Closed_infinite_broomhttps://en.wikipedia.org/wiki/Infinite_broomhttps://en.wikipedia.org/wiki/Topologist%2527s_sine_curvehttps://en.wikipedia.org/wiki/Topologist%2527s_sine_curvehttps://en.wikipedia.org/wiki/Topologist%2527s_sine_curvehttps://en.wikipedia.org/wiki/Novak_spacehttps://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactificationhttps://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactificationhttps://en.wikipedia.org/wiki/Box_topology


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