# 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].

22
questions

**4**

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**1**answer

190 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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97 views

### 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 ...

**1**

vote

**1**answer

43 views

### 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 ...

**0**

votes

**1**answer

255 views

### 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 ...

**5**

votes

**1**answer

180 views

### 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

664 views

### 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 ...

**0**

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**0**answers

94 views

### 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 ...

**2**

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**0**answers

57 views

### 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 ...

**2**

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233 views

### 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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238 views

### 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,...

**11**

votes

**1**answer

1k views

### 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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votes

**1**answer

995 views

### 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 ...

**2**

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**1**answer

229 views

### 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 ...

**4**

votes

**1**answer

351 views

### 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)...

**9**

votes

**4**answers

3k views

### 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 ...

**3**

votes

**1**answer

521 views

### 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(...

**5**

votes

**3**answers

2k views

### Discrete subspaces of Hausdorff spaces

does every infinite hausdorff space contains a countable infinite discrete subspace?

**3**

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**4**answers

716 views

### 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 ...

**10**

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**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 ...

**6**

votes

**2**answers

800 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 ...

**72**

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.
...

**1**

vote

**10**answers

6k 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 ...