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Recall a Polish space is a completely metrizable separable space. Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous m …

I am reasking a year-old math.stackexchange.com question asked by someone else. (For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.) The Tietze ex …

Yes, $\mathbb X$ is a Polish space, even a computable one, but it doesn't look like it has a nice metric. (I figured out the answer on my own question, but any other insightful answers or references …

If $X$ and $Y$ are Polish and $Y$ is compact, then YES. (I think my proof can be fixed to handle the noncompact setting, but I don't see how right now.) My proof involves this lemma. Lemma. Assume …

Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-Prok …

Recall a topological space $X$ is path connected if for all $x,y \in X$ there is a continuous function $f\colon [0,1] \to X$ such that $f(0)=x$ and $f(1)=y$. Say that $X$ is continuously path connect …

This is the first in a pair of questions. For the other see here. Dustin Clausen and Peter Scholze have a theory of condensed sets, which is a slightly different take on topology. For most cases, th …

This is the second in a pair of questions. For the other see Are representations in computable analysis the equivalent to countably-generated condensed sets?. Dustin Clausen and Peter Scholze have a …

I know in computable analysis, which is closely related to the descriptive set theoretic questions you are asking, that the Lévy–Prokhorov metric on the weak topology is useful. (I think the narrow t …