Questions tagged [hausdorff-spaces]

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].

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Why does normality imply that a countable base $B$ contains at least one set $U$ whose closure is a subset of another set $V \in B$?

I'm reading Aliprantis and Border's excellent text, Infinite Dimensional Analysis: A Hitchhiker's Guide (PDF available at link, assuming I've done this properly), and I've reached an impasse in the ...
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Subsets of the Cantor set

A copy of the Cantor set is a space homeomorphic to $2^{\omega}$. Suppose that $X$ is a Hausdorff space that contains a copy $C^{\prime}$ of the Cantor set. Let $U$ be a nonempty subset open in $C^{\...
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2 votes
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A question about infinite product of Baire and meager spaces

Proposition 1: For any space $X$ and an infinite cardinal $\kappa$, the product $X^{\kappa}$ is either meager or a Baire space. Does anyone have any suggestions to demonstrate Proposition 1? I was ...
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6 votes
1 answer
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Is a closed subset of an extremally disconnected set again extremally disconnected?

Let $T$ be a compact Hausdorff extremally disconnected set (so $T$ is a compact Hausdorff space, such that the closure of each open subset is again open). Let $S \subseteq T$ be a closed subset. ...
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1 vote
1 answer
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Spaces whose interiors of retracts is a base of the topology

Definition:   topological space $\ X\ $ is   r-basic $\ \Leftarrow:\Rightarrow\ $ the interiors of retracts of $\ X\ $ form a topological base of $\ X.$ Main question: Are r-basic spaces mentioned in ...
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1 answer
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extending disjoint open subsets of a normal Hausdorff space

Under what assumptions on $C$ and $X$ is the following true ? I was neither able to find a counterxample or prove this, though it appears that compactness, e.g. assuming $X$ is compactly generated, ...
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5 votes
1 answer
253 views

Is the tensor product of compactly generated Hausdorff abelian groups again Hausdorff?

Consider the tensor product $G \otimes_{\mathbb{Z}} H$ of two abelian groups $G$ and $H$. If $G$ and $H$ are topological groups, we can give $G \otimes_{\mathbb{Z}} H$ a topology as follows. For any $...
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1 vote
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Subspaces of compact spaces and quotients of Hausdorff spaces

Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following ...
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1 vote
1 answer
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Hausdorff quotient collapsing and separating a prescribed collection of disjoint closed subsets

Let $X$ be a compact Hausdorff space (I don't mind assuming it's metrizable). Let $A_i$ $i\in \mathbb{N}$ be a collection of disjoint closed subsets of $X$. My question: Does there exist a Hausdorff ...
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Countable intersections in topological space

If a T1 topological space is closed under countable intersections, does this necessarily make the topology discrete? It is easy to construct a counterexample if the topological space is not assumed to ...
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6 votes
1 answer
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Uniqueness of limits and compactness implies closure

It is not difficult to prove that in a Hausdorff topological space every compact set is closed, and almost trivial that if in a topological space X every compact set is closed then X is T1. As ...
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4 answers
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Is a separable compact Hausdorff space already metrizable? [closed]

It is a known fact that a 2nd countable compact Hausdorff space is metrizable. What if we weaken the 2nd countable to separable only - is the space still metrizable? The core of the question, or a ...
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A relative version of Urysohn's Lemma?

Let $f:Y\to X$ be a continuous surjective map between locally compact Hausdorff spaces. Assume there is a continuous section $s:X\to Y$ which has closed image and is a homeomorphism to the image. I ...
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2 votes
0 answers
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Breaking down the co power of a topological space

Consider a compact, Hausdorff topological space which is homeomorphic to its own co-power over an index set $I$, so $X \cong \prod_{i \in I } X$. Is there necessarily another topological space, which ...
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The Cauchy Problem in General Relativity: Existence of a Hausdorff Development

This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...
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Relationship between weak Lp and strong Lq topologies for q<p

Specificaly: Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence? Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not,...
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11 votes
1 answer
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Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?

Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity. Definition 1: A triplet $ (\Sigma,h,k) $ is called an initial data set if $ (\Sigma,...
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1 answer
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If X is a Haussdorf topological space and R and equivalence relation on X, when is X/R Haussdorf?

I was wondering if there are some necessary and sufficient conditions for the quotient space to be Haussdorf. I have been trying a little for a while, but I only got very restrictive sufficient ...
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1 answer
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Relative extremely disconnected space

A topological space $X$ is called relative extremely disconnected if it has a base $B$ (for open subsets) such that disjoint elements in $B$ have disjoint closure. Does it exist an infinite ...
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4 votes
1 answer
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Is normality of a Hausdorff space consequence of some property of open domains?

Let $\mathrm{r}\mathscr{O}$ be the family of open domains (regular open sets) of a topological space $\langle X,\mathscr{O}\rangle$, that is: $$A\in\mathrm{r}\mathscr{O}\iff A=\mathrm{int(\mathrm{cl(A)...
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9 votes
4 answers
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Examples for "nice" Boolean algebras that are not complete or not atomic

A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms). Boolean Algebras that are complete as ...
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3 votes
1 answer
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Jet spaces between non Hausdorff manifolds

I found it very hard to find literature about smooth manifolds that are not required to be Hausdorff. In particular I'm interested in their local properties: 1.) Are the $r$-th order jet bundles $J^r(...
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6 votes
3 answers
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Discrete subspaces of Hausdorff spaces

does every infinite hausdorff space contains a countable infinite discrete subspace?
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3 votes
4 answers
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Does countable compactness imply local compactness in Hausdorff spaces?

The question arose while comparing the notions of compactness, countable compactness, local compactness, and "Lindelofness" in Hausdorff spaces. It is straightforward to show that compactness implies ...
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11 votes
3 answers
1k views

Minimal Hausdorff

A Hausdorff space $(X,\tau)$ is said to be minimal Hausdorff if for each topology $\tau' \subseteq \tau$ with $\tau' \neq \tau$ the space $(X,\tau')$ is not Hausdorff. Every compact Hausdorff space ...
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6 votes
2 answers
901 views

Compact cover of a Hausdorff compact space

In his book "Riemannian Geometry" do Carmo cites the Hopf-Rinow theorem in chapter 7. (theorem 2.8). One of the equivalences there deals with the cover of the manifold using nested sequence of compact ...
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74 votes
5 answers
5k views

How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. ...
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0 votes
10 answers
8k views

What is an explicit example of a sequence converging to two different points? [closed]

In principle a sequence in a non-Hausdorff space can converge to two points simultaneously. Can anyone give me an explicit example of the above? Or tell me any method of generating such kinds of ...
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