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There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter. But there's ...

Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914. I wonder if the following question of ...

This is a reformulation of this MO question which recieved little or no attention due to the fact that the OP gave no motivation whatsoever. I found the question quite interesting and decided to give ...

Consider a set $X\subseteq \mathbb{R}$ such that $X$ is not separable wrt its subspace topology For all $r\in\mathbb{R}$ there exists a sequence $(x_n)_{n\in\omega} \subset X$ converging to $r$ In a ...

A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have ...