Cardinal function

Remarks

Common examples, for a space $X$:

1. the weight $w(X)$, the least cardinality of a base of the topology;
2. the density $d(X)$, the least cardinality of a dense subset;
3. the character $\chi(X)$, the least cardinal $\kappa$ such that every point has a neighbourhood base of cardinality at most $\kappa$;
4. the cellularity $c(X)$, the supremum of the cardinalities of families of pairwise disjoint nonempty open sets.

Many authors add $\aleph_0$ to each value, so that these functions are infinite. Two basic inequalities follow. First, $d(X) \le w(X)$: choose one point in each nonempty member of a base. Second, $\abs{X} \le 2^{w(X)}$ for a T0 space $X$, since a point is determined by the base members that contain it.

Example: for $X = \R$, one has $w(X) = d(X) = \aleph_0$, by the base of intervals with rational endpoints and the dense set $\Q$.

Non-example: the rule $X \mapsto$ the number of points of $X$ that lie in $[0, 1]$, defined only for subspaces of $\R$, is not a cardinal function. It is not invariant under homeomorphism: $[0, 1]$ and $[2, 3]$ are homeomorphic but give $\mathfrak c$ and $0$.