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We call $X$ strongly collectionwise normal if the set $\mathcal{U}_\Delta$ of all neighbourhoods of the diagonal $\Delta_X$ of $X\times X$ is a uniformity. This is equivalent to the property that for …

Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications: Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ locall …

I am looking for where this specific theorem of Blair is originally located: Theorem. Let $S\subseteq X$, the following are equivalent: $S$ is $z$-embedded If $A, B\subseteq S$ are disjoint zero-set …

Recently I've proven the following theorem Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent: Every compactification of $X$ is zero-dimensional …

I am looking for locally compact Hausdorff spaces $X$ with the following property: If $f:X\to Y$ is a perfect map onto locally compact Hausdorff space $Y$, then $Y$ is zero-dimensional. One can see …

Let $(P_1)$ be the property: Every locally finite open cover of $X$ has finite subcover. Let $(P_2)$ be the property: Every locally finite open cover of $X$ is finite. Let $(P_3)$ be the property: Eve …

In the article A Perfectly Normal, Locally Compact, Noncollectionwise Normal Space Form $\lozenge^\ast$ by Daniels and Gruenhage (I presume "form" is a typo and it should be "from", but it's like that …

Let $X$ be a uniform space, $S\subseteq X$ and $f:S\to \mathbb{R}$ bounded uniformly continuous, then there exists a uniformly continuous extension of $f$ to $X$. (Katětov, 1951) I am looking for pr …

According to @Tyrone the term well-embedded set was first used in Measures on Metacompact Spaces by W. Moran. In the article Extensions of Zero-sets and of Real-valued Functions by R. Blair and A. Hag …

I was browsing Gillman and Jerison for known relations between zero-sets, $C$-embedded sets and so on. The spaces I'm considering are $T_{3.5}$. There are two properties that pseudocompact spaces have …

Note: What I call a measurable cardinal seems to be non-standard among set theorists, and should be called a $\sigma$-measurable cardinal. I know that a discrete space is realcompact iff its non-measu …

Consider the ring $C = C(X) = C(X; \mathbb{R})$ of continuous functions $f:X\to \mathbb{R}$ where $X$ is a Tychonoff space. This is naturally a lattice ordered ring by setting $f\geq 0$ iff $f(x)\geq …

Let $w(X) = \inf\{|\mathcal{B}| : \mathcal{B} \text{ is a base for }X\}$ be the weight of topological space $X$. For metric spaces and locally compact spaces we have inequality $w(X)\leq |X|$. This in …

We work in ZFC. Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not isomorphic to $\mathbb{R}$. A field $E$ is ca …

I'm learning about topological groups from Arhangelskii and Tkachenko "Topological groups and related structures" and this is one of the exercises there. A paratopological group is a group with topolo …