Freedom Math Dance

The first edition of Bourbaki's General Topology (chapter I, §9, p. 56) contains the following theorem. Proposition 3. Soient $E$, $F$...

The axiom of choice says that an arbitrary product $\prod_{i\in I} A_i$ of non-empty sets $A_i$ indexed by a set $I$ is non-empty. It is wel...

This is the title of a very short paper by Steven Kleiman published in L'enseignement mathématique , and which should be studied by eve...

In the book Categories and sheaves by Kashiwara and Shapira, I found a nice argument for the fact that the category of sets is not equivale...

Let me recall the statement of this theorem. Theorem. Let $X$ be a topological space, let $U,V$ be connected open subsets of $X$ such t...

I thought a little more about the proof that an ultraproduct of finite sets is either finite (if the family of cardinals is bounded with res...

Ultrafilters won't make american coffee better, but they have nice applications in mathematics. In fact, the last three (working) days,...