138:
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184:
are finite subsets, all compact subsets are closed, another condition usually related to the
Hausdorff separation axiom.
251:
176:. However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique. Since
298:
133:{\displaystyle {\mathcal {T}}=\{A\subseteq X:A=\varnothing {\mbox{ or }}X\setminus A{\mbox{ is countable}}\}.}
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is an uncountable set then any two nonempty open sets intersect, hence the space is not
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191:. The cocountable topology on an uncountable set is
187:The cocountable topology on a countable set is the
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52:. It follows that the only closed subsets are
259:reprint of 1978 ed.), Berlin, New York:
8:
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60:. Symbolically, one writes the topology as
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154:omits only countably many points of
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165:, as all singletons are closed.
146:with the cocountable topology is
56:and the countable subsets of
1:
22:countable complement topology
252:Counterexamples in Topology
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40:, that is all sets whose
209:weakly countably compact
150:, since every nonempty
247:Seebach, J. Arthur Jr.
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215:, hence not compact.
213:countably metacompact
135:
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18:cocountable topology
243:Steen, Lynn Arthur
230:List of topologies
130:
122:
120: is countable
106:
270:978-0-486-68735-3
225:Cofinite topology
201:locally connected
189:discrete topology
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105:
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299:General topology
283:(See example 20)
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28:consists of the
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261:Springer-Verlag
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205:pseudocompact
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158:. It is also
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178:compact sets
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25:
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36:subsets of
34:cocountable
24:on any set
236:References
142:Every set
42:complement
249:(1995) ,
197:connected
174:Hausdorff
112:∖
99:∅
84:⊆
50:countable
30:empty set
293:Category
219:See also
152:open set
148:Lindelöf
32:and all
279:0507446
195:, thus
277:
267:
257:Dover
265:ISBN
211:nor
203:and
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180:in
168:If
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