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According to https://mathoverflow.net/a/8994/4600, $\log_2$ of the number of topologies on $n$ elements is $\sim n^2/4$. So let's say there are $2^{n^2/4}$ topologies. Then the number of topologies on …

This is false. Let $U=\mathbb N\times\mathbb N$ and let $S$ consist of all columns. Let $c$ be a product ultrafilter, i.e. a set is large if most of its intersections with columns are large. Then no e …

According to https://arxiv.org/abs/1210.1008 Example 2(c)... yes, they are all homeomorphic to $\mathbb Q$!

Couldn't read the Russian image text but I'll take a stab at it: If there are no restrictions on $M\mapsto\overline M$ at all then it's just a unary set operation: a function from the power set of $R …

Inspired by Hausdorff dimension, we can try to let $X$ consist of numbers whose decimal expansion is of the form $$ 0.x_100x_4x_5x_60000\dots $$ and $Y$ consist of the "complementary" numbers: $$ 0.0x …

Actually, the converse implication doesn't hold. Let $X$ and $Y$ be posets with greatest elements $1_X$, $1_Y$, and let $\delta_x$ and $\delta_y$ be the constant $1_X$, $1_Y$ maps. These are clearly …

No, there is no such set. The situation for meager sets is dual to that described by Pietro Majer in a comment on Translates of null sets, "I was vaguely thinking to Hausdorff measures w.r.to gaug …

No, let $X$ be the set of those irrationals in $x\in (0,1)$ with binary expansion $$x=0.x_1x_2\dots$$ such that if we define $x^{\text{even}}, x^{\text{odd}}$ by $$x^{\text{even}}=0.x_2x_4x_6\dots$$ $ …

Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is an open set $U$ with $D=S\cap U$? I'm asking merely out of curiosity, but I'll mention that thi …

Suppose $f:C\to C$ is a homeomorphism, where $C=\{0,1\}^{\mathbb N}$ is Cantor space. Suppose $f$ preserves $=^*$ (equality on all but finitely many coordinates). Does it follow that $f$ also reflects …

Infinite, complete, separable linear order with at most countably many jumps and not both a greatest and least element. See http://www.math.uni-hamburg.de/home/geschke/papers/SeparableLinearOrders2. …

The set $\mathbb R\setminus\mathbb Q$ of all irrational numbers.

What conditions can we impose on $T$ that guarantee $[T]$ contains a 1-generic member? An element that is 1-generic relative to $T$ will not be on $[T]$ unless $[T]$ contains a whole clopen cone $[\ …

In philosophy, this would be called family resemblance -- if $E_i\cap E_j\ne\emptyset$ and $E_j\cap E_k\ne\emptyset$ then $E_i$ and $E_k$ have a family resemblance. That is, perhaps I have no common …

According to Wikipedia, which first talks about the other not-finite-sounding notions, However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century …