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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 …

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 …

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 …

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 …

Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set. Is there a always a compact connected $L\subset X$ such that $K\subset L$? Th …

I am looking for any information about the following property for a compact Hausdorff space $K$: For any sequence $\left(U_{n}\right)$ of nonempty open sets (not necessarily distinct) there is a disjo …

Recall that a subset $A$ of a Boolean algebra $B$ is a $\pi$-base if for every $b>0$ there is $a\in A$ with $0<a\le b$. For example, the definition of atomicity says that atoms constitute a $\pi$-base …

Recall that a topological space is extremally disconnected if the closure of any open set is open. A continuous map is quasi-open if it maps nonempty open sets onto sets with nonempty interior. For so …

Originally asked on MSE. Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ end …

Let $X\subset \mathbb{R}^2$ be open connected (and let's say bounded), let $x\in X$ and $y\in\partial X$ be such that there is a Jordan curve $\gamma:[0,1]\to X\cup\{y\}$ such that $\gamma(0)=x$ and $ …

Let $E$ be a locally convex topological vector space over $\mathbb{R}$. The projectivization $PE$ is the quotient of $E\backslash\{0_{E}\}$ with respect to the equivalence relation $e\sim f$ if $e=\la …

Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$. What can be said abo …