# Questions tagged [hausdorff-spaces]

The hausdorff-spaces tag has no usage guidance.

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Discrete subspaces of Hausdorff spaces

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

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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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### 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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### 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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### 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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### 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 ...