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If $[X]^2$ is compact then $X$ is finite. Let $\Delta \subseteq X^2$ be the diagonal. It is easy to show that the map $X^2 \setminus \Delta \to [X]^2$ given by $(x,y) \mapsto \{x,y\}$ is continuous, …

It is easy to see that the operation $X \mapsto [X]^2$ preserves the properties: countable, second-countable, regular, no isolated points. Hence $\mathbb{Q} \cong [\mathbb{Q}]^2 $.

The answer is no. It was proved by Mary Ellen Rudin in $\aleph$-Dowker spaces (1978) that for any non-discrete Hausdorff space $S$ there is a normal Hausdorff space $N$ such that $S\times N$ is not no …

According to this article of Colasante and Van Der Zypen, it is an open problem to characterize those reflexive and symmetric relations for which there is such a topology. If we restrict ourselves to …

The answer is no, by the argument given by Henno in his comment. It true that there are exactly $2^{2^\kappa}$-many non-homeomorphic topologies on a set of size $\kappa \geq \aleph_0$. See for example …

A space $X$ is said to have countable tightness if whenever $A \subseteq X$ and $p\in \bar{A}$, there is a countable $B \subseteq A$ such that $p \in \bar{B}$. It is not hard to see that a space has c …

Yes, see for example "Continua which admit only the identity mapping onto non-degenerate subcontinua" by H. Cook (Fund. Math. 60, 1967, 241-249).

This is an attempt to formalize Christian Remling´s idea. Cover $K$ by finitely many balls $\{B(x,r_x): x \in S \}$ such that $\overline{B(x,2r_x)} \subseteq U_x$. Let $\delta$ be the minimum of the …

A space is $T_{1/2}$ if every singleton is either open or closed. Sierpinski's space is an example of a $T_{1/2}$ not $T_1$ space. If $X$ is a $T_{1/2}$ space, let $A$ be the set of non-isolated point …

It was proved independently by Efimov and Gerlits that the answer is yes if $\kappa$ has uncountable cofinality. In fact they proved this for any dyadic $X$ (it is well known that Dugundji compacta ar …

No. If $S$ is a maximal discrete subset of a $T_1$-space $X$, then every point of $S$ is isolated in $X$ (in fact, $S$ must be the set of isolated points of $X$ and it must be dense in $X$). Thus if $ …

First note that in a space with countable tightness: $P$-point $\Leftrightarrow$ weak $P$-point $\Leftrightarrow$ Isolated point. So you are looking for a point for which every deleted neighborhood c …

It was shown by A.K. Misra in "A topological view of $P$-spaces" that any regular $\aleph_1$-Lindelöf $P$-space is normal (see Corollary 4.6). Also, it is not hard to see that any linearly Lindelöf sp …

The boolean algebra of clopen subsets of a space satistying conditions 1,2 and 3 is always a base for the topology (i.e. the space is zero-dimensional). So the space is homeomorphic to the Stone space …

It is not true that if $\mathscr{F}$ is dominating then $\{K_f:f\in\mathscr{F}\}$ is a cover of $\omega^\omega$. For example it is possible to have a dominating $\mathscr{F}$ such that $f(4)=6$ for ev …