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Not necessarily, because in some cases of $X$ Hausdorff, compact, infinite, $[X]^2$ is isomorphic to an open subset of $X\times X$ that is not closed (therefore not compact). Here's an example. Let $ …

This (rather exotic) covering property is often called "Property D". Here you find a nice survey of $D$-spaces. On p. 11, assuming $\textrm{(CH)}$, a space $Y$ is mentioned that has property $D$, but …

Is there an example of a family $(X_i)_{i\in I}$ of metacompact spaces, such that their box product $\prod_{i\in I}^{\textrm{Box}}X_i$ is not metacompact?

The answer is Yes. Suppose $(X,\tau)$ is $R_1$ and compact. Pick $V$ open and let $x\in V$. For each $y\in X\setminus V$ there are open neighborhoods $U(y)$ of $x$ and $V(y)$ of $y$ such that $U(y)\c …

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$. A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we …

Let $\kappa$ be an infinite cardinal. Does there exist a topology $\tau_{\kappa+1}$ on $\kappa+1$ such that for any topological space $(X,\tau)$ with $|X|=\kappa$ the following statement is true? …

Yes - the correspondence $M^p \mapsto M^{*p}$ (and vice versa) is exactly what you are looking for.

Given a set $X$, is there a topology $\tau$ such that the identity $\text{id}_X$ on $X$ is the only continuous injective self-map? (This is Joel David Hamkins's recent question in the category $\math …

For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$ Clearly, $R_{im}(X,\tau)$ is reflexive, and …

For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$ Clearly, $R_{im}(X,\tau)$ is reflexive. This …

Is there an infinite topological space $(X,\tau)$ with the following property? There is an open cover ${\cal U}^*$ such that $X\notin {\cal U}^*$; every finite subset $F\subseteq X$ is contained in …

Let $(X,\tau)$ be a topological space. We define the "moving" relation by setting $$ x \simeq_m y \text{ iff there is a homemomorphism }\varphi: X\to X \text{ such that } \varphi(x) = y.$$ Clearly $\ …

Let $(X,\tau), (Y,\sigma)$ be topological spaces with $|X|$ infinite and suppose $\varphi:X\to Y$ is a bijection such that for all $x\in X$ we have that $(X\setminus\{x\}) \cong (Y\setminus\{\varphi(x …

Given a topological space $C$ and points $c_0, c1\in C$ we say that a topological space $(X,\tau)$ is $(C,c_0,c_1)$-connected if and only if for all $x,y\in X$ there is $f:C\to X$ continuous with $f(c …

A topological space $(X,\tau)$ is connected if and only if the only continuous maps $f:X\to\{0,1\}$ (where $\{0,1\}$ carries the discrete topology) are the constant maps. Are there other examples of …