In ZF the following are equivalent: (1) ℕ is a Lindelöf space. (2) ℚ is a Lindelöf space. (3) ℝ is a Lindelöf space. (4) The axiom of countable choice holds for subsets of ℝ.

[Papanikolaou 1997]: If the Euler–Mascheroni constant 𝛾 is rational, then its denominator exceeds 10²⁴⁴⁶⁶³.

Let M̅_g,n be the Deligne–Mumford–Knudsen compactification of the moduli space M_g,n of curves of genus g with n marked points. [Canning–Larson–Payne 2023]. Given g ≥ 1, n ≥ 0, TFAE: (1) H¹¹(M̅_g,n; ℚ) ≠ 0. (2) g=1 and n≥11.

22⋅arctan(24478/873121) + 17⋅arctan(685601/69049993) equals 𝜋/4.

It follows that the topological version of the Whitney embedding theorem holds for compact (connected) n-manifolds: they all embed in ℝ²ⁿ. The non-compact ones are also believed to embed as such, but this does not seem to be written anywhere.

For a field k and a (finite) set X of variables, let k((X)) be the field of fractions of the formal power series ring k[[X]]. Theorem. There is a chain C of subfields within the field extension k((x,y)) → k((x))((y)) with |C| = |ℝ|.

>17 year old trans girl has article written about her mathematical contributions >68 year old creep sexologist (redundant adjective) complains about the article not sexualising her trans women truly are the women of women

A set X is called amorphous if X is infinite and whenever X = Y ⊔ Z, either Y or Z is finite. Theorem. ZF has models in which there are amorphous sets.

Given any year in the commonly used Gregorian calendar, • the last day of February, • April 4 (4/4), • June 6 (6/6), • August 8 (8/8), • October 10 (10/10), • December 12 (12/12) all occur on the same day of the week. For 2025 this so-called Doomsday is Friday.

A short proof of Cantor–Bernstein–Schröder: chiasme.wordpress.com/2013/11/20/a-s…

The upper bound is improved to 46 here: arxiv.org/abs/2409.15709

Every position of Rubik's cube can be solved in at most 20 half-turn moves. Moreover there are positions for which 19 moves is not enough.

This position with white to move is impossible to reach within the current laws of chess. Source: anselan.com/tutorial.html

[Fontaine 1985]. There is no nontrivial abelian scheme over ℤ.

[Davis 1955]. Given a lattice (L, ≤), TFAE: (1) Every monotone f:L→L has a fixed point. (2) L is complete.