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Results tagged with hausdorff-spaces
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user 5903
Use this tag for questions that specifically address the role of the Hausdorff (T_2) condition, or about the set of Hausdorff topologies, etc. For a topological question with the Hausdorff assumption, just use [gn.general-topology].
8
votes
Accepted
Does countable compactness imply local compactness in Hausdorff spaces?
Examples abound: take for instance a $\Sigma$-product of two-point spaces. To be specific let $X$ be the set of points in $\lbrace0,1\rbrace^{\omega_1}$ that have only countably many coordinates that …
- 11.4k
2
votes
Accepted
extending disjoint open subsets of a normal Hausdorff space
The first is true in any hereditarily normal space: separated sets have disjoint neighbourhoods.
It fails in the compact product $(\omega_1+1)\times(\omega+1)$ (Tychonoff's plank with corner point). T …
- 11.4k
5
votes
Spaces whose interiors of retracts is a base of the topology
For a general source of counterexamples: look at connected but not locally connected spaces. The retracts are connected but the neighbourhoods of some points are not.
The Topologist's sine curve, Knas …
- 11.4k
3
votes
Accepted
Subsets of the Cantor set
To answer the question: every point of $U$ is an accumulation point of $D\cap U$, hence there are continuum many accumulation points. The ambient space $X$ plays no role here; everything takes place i …
- 11.4k