The meagre subsets of a topological space form a sigma-ideal

Given: The objects, hypotheses, and choice principles stated above.

Let $X$ be a topological space and let $A\subseteq X$. The set $A$ is nowhere dense when $\mathrm{int}(\bar{A})=\mathrm{\varnothing }$ (def-interior-closure-boundary-top). It is meagre when there is a sequence $(N_n{)}_{n\in \mathbb{N}}$ of nowhere dense subsets of $X$ with $A\subseteq {\bigcup }_n{N}_n$. It is residual, or comeagre, when $X\setminus A$ is meagre. The empty union shows that $\mathrm{\varnothing }$ is meagre, including when $X=\mathrm{\varnothing }$. (Nowhere dense, meagre, residual, and comeagre subsets of a topological space).

$\mathbb{N}\times \mathbb{N}\approx \mathbb{N}$ (def-equinumerous): the plane of pairs of naturals is countably infinite (def-countable). The bijection is exhibited, not merely asserted to exist. Define $2^m$ by recursion on $m$ (thm-recursion) by $2^0=1$ and $2^{\sigma (m)}=2^m+2^m$, and set $J(m,n)=2^m\cdot \sigma (n+n),\text{that is}J(m,n)=2^m(2n+1)\mathrm{.}$ Then $J$ is a bijection from $\mathbb{N}\times \mathbb{N}$ onto $\mathbb{N}\setminus \{0\}$, and $\sigma$ is a bijection from $\mathbb{N}$ onto $\mathbb{N}\setminus \{0\}$, so ${\sigma }^{-1}\circ J$ is a bijection $\mathbb{N}\times \mathbb{N}\to \mathbb{N}$. What makes $J$ bijective is the decomposition of a nonzero natural into a power of two times an odd number, existence and uniqueness both. ($\mathbb{N}\times \mathbb{N}\approx \mathbb{N}$).

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) selects one member from each nonempty set in a sequence of sets. It is used below for the sets of nowhere dense covering sequences.