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  "body": "\n## Definition\n\nA *cardinal function* is a rule $\\varphi$ that assigns to each [[topological space|0CQHFPGJ45]] $X$ a [[cardinal number|0EBACQFAZW]] $\\varphi(X)$.\nIt is required that $\\varphi(X) = \\varphi(Y)$ whenever $X$ and $Y$ are [[homeomorphic|0QT00PB5G2]].\n\n## Remarks\n\nCommon examples, for a space $X$:\n\n1. the weight $w(X)$, the least [[cardinality|0G0ASKNA6Z]] of a [[base|0NA0732YAG]] of the topology;\n2. the density $d(X)$, the least cardinality of a [[dense|0BSJSX7ME2]] subset;\n3. the character $\\chi(X)$, the least cardinal $\\kappa$ such that every point has a neighbourhood base of cardinality at most $\\kappa$;\n4. the cellularity $c(X)$, the supremum of the cardinalities of families of pairwise disjoint nonempty open sets.\n\nMany authors add $\\aleph_0$ to each value, so that these functions are infinite.\nTwo basic inequalities follow.\nFirst, $d(X) \\le w(X)$: choose one point in each nonempty member of a base.\nSecond, $\\abs{X} \\le 2^{w(X)}$ for a [[T0 space|07SG12TQH3]] $X$, since a point is determined by the base members that contain it.\n\nExample: for $X = \\R$, one has $w(X) = d(X) = \\aleph_0$, by the base of intervals with rational endpoints and the dense set $\\Q$.\n\nNon-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.\nIt is not invariant under homeomorphism: $[0, 1]$ and $[2, 3]$ are homeomorphic but give $\\mathfrak c$ and $0$.\n",
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