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The Sorgenfrey plane is not metacompact by Example 2 in these notes. However, the Sorgenfrey plane is the product of two paracompact (and hence metacompact) spaces. Therefore metacompact spaces are no …

There are even non-normal separable complex manifolds. The paper Complex analytic manifolds without countable base by Calabi and Rosenlicht gives examples $M,S$ of a non-normal separable complex manif …

Let me answer this question by assuming that by "stationary" you mean "eventually constant." One may characterize the topological spaces where every convergent sequence is eventually constant in term …

A simple such characterization exists when one is working with proximity spaces instead of topological spaces. Suppose that $(X,\delta)$ is a proximity space with complete containment relation $\ll$. …

Suppose that $(X,\mathcal{T})$ is a topological space. Then define a relation $\prec$ on $\mathcal{T}$ by letting $U\prec V$ if and only if $\overline{U}\subseteq V$. Define a relation $\ll$ on $\math …

By the following proposition, the only Hausdorff spaces which are completely determined by their dense subsets are the discrete spaces. Proposition. Let $(X,\mathcal{T})$ be a $T_{0}$ topological spa …

Let me give a proof that makes use of the ordering inherited from $\mathbb{R}$. Suppose that $A\subseteq\mathbb{R}$ is a countable compact space. Then $A$ has no subset $B$ order isomorphic to the ra …

I claim that every finite $T_{0}$ critical space is isomorphic to $(n,\tau_{n})$. Suppose that $X$ is a finite $T_{0}$-space. Let $\leq$ be the specialization ordering on $X$. Then $U\subseteq X$ is …

$\textbf{Finite $T_{0}$-spaces are $T_{inj}$}$ I claim that every finite $T_{0}$-space is $T_{inj}$. If $(X,\tau)$ is a finite $T_{0}$-topology, then there exists a partial ordering $\leq$ on $X$ so …

I claim that for each cardinal $\lambda$, there is a connected space $C$ and $c_{0},c_{1}\in C$ such that whenever $|X|<\lambda$, then $X$ is connected if and only if for all $x,y\in X$ there is some …

The uniform continuity property that you have mentioned does guarantee that $X$ is a retract of the hyperspace $H(X)$. Let $[X]^{<\omega}$ denote the collection of all finite subsets of $X$. Then $[X] …

I claim that every paracompact locally compact totally disconnected space can be written as a disjoint union of compact open sets. Every paracompact locally compact space can be partitioned into a co …

As stated in the comments, one can show that $\omega_{1}$ is not paracompact hence non-metrizable. However, one can also give a direct proof that $\omega_{1}$ is not metrizable without resorting to pa …

In this answer, I will assume that all spaces are Hausdorff. It is well known that a space is compact if and only if it is pseudocompact and realcompact[See The Stone Cech Compactification by Russel W …

David White's result may be generalized to necessary and sufficient conditions for when a locally compact space is $\sigma$-compact. A locally compact space is $\sigma$-compact if and only if it is Li …