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I think the answer is no. I'll basically point to two exercises in Engelking. A map $q:X\to Y$ is called hereditary quotient, if for any $B\subset Y$, the restriction of $q$ on $q^{-1}(B)$ is a quoti …

I was looking for a related fact, and surprisingly couldn't find anything relevant, except of this question. Even though it was answered 10 years ago, perhaps the following result could be useful to s …

This is an answer to the updated question. Proposition: If a closed $S\subset [0,1]\times [0,1]$ intersects every connected set with a full projection onto the $x$-axis, then it has a component with a …

Let $X$ and $Y$ be Tychonoff (i.e. completely regular Hausdorff) topological spaces and let $\varphi:X\to Y$ be a continuous surjection that also has a property that $\operatorname{int}\overline{\varp …

Let $K$ be compact Hausdorff, let $U\subset K$ be open and dense, and let $x\in K\backslash U$. Can we find a disjoint collection $\{U_i,~ i\in I\}$ of open subsets of $U$ and a collection $\{K_i,~ i\ …

Let $U$ be a regular open set in a Tychonoff space $X$ (regular means that it is an interior of a closed set). [ In my specific situation $U$ is of the form $\operatorname{int} f^{-1}(0)$, where $f$ i …

Here is an excerpt from a mathscinet review of the paper Johannes de Groot - Topological characterization of metrizable cubes (1972): A connected $T_1$ space of dimension $n=1,2,⋯,\infty$ is homeo …

Recall that a topological space is called Fréchet–Urysohn if every convergent net contains (as a set) a sequence, which is convergent to the same limit. I want to refine this property as follows. Let …

In general topology there is two ways of introducing a topology on the space of (continuous) maps between, say, metric spaces: set-open topology and uniform topology (it is a uniformity of uniform con …

In case, anyone is interested, my question happened to be rather simple. Set-open topology coincides with the uniform topology if the target space is homogeneous enough. More precisely: Let $X$ be a …

Let $X$ be a topological space. $Y$ is a discrete subset of $X$ if it has a discrete topology induced by the topology of $X$. This is equivalent to the fact that for every $y\in Y$ there is an open $U …

See Corollary 2.8 in this paper: If $X$ is perfect, compact and metrizable, then there is a non-atomic regular Borel measure of full support on $X$.

This is meant to fill in some of the details outlined by Anton Petrunin's answer, and also to refine the statement slightly. Recall that a compact connected Hausdorff space is called a continuum. We …

Here is another answer, based again on Anton Petrunin's idea, but obtaining a slightly different result. Theorem. Let $X$ be a connected and locally path-connected completely metrizable space. Then f …

The title says it all. Let $A$ be a path connected $F_\sigma$ subset of a plane (or more generally $\mathbb{R}^n$). Recall that a subset is called $F_\sigma$ if it is a union of a sequence of closed s …