Under choice, weight $w(X)$, density $d(X)$, local character $\chi(x,X)$, and character $\chi(X)$ as raw cardinal minima and a supremum

Assume the Axiom of Choice (The Axiom of Choice) and let $X$ be a topological space. The weight $w(X)$ is the least cardinality of a basis for $X$, and the density $d(X)$ is the least cardinality of a dense subset of $X$ (Basis and subbasis for a topology, and the topology generated by a family of sets, Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets, Cardinal (initial ordinal) and cardinality).

For $x\in X$, the local character $\chi(x,X)$ is the least cardinality of a neighbourhood base at $x$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open). The character is the raw cardinal supremum $\chi(X)=\sup\{\chi(x,X):x\in X\}.$

No $\aleph_0$ normalization is imposed. In particular a one-member local base has cardinality $1$, not $\aleph_0$. The forward lemmas named in justified_by establish the asserted minima and supremum.