# Questions tagged [hausdorff-spaces]

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18 questions
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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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### 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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### 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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### 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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### 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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### 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,...
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 ...