---
tag: XVNJPJ53W8
kind: concept
name: Cardinal function
msc: [54A25]
sources:
  - url: https://en.wikipedia.org/wiki/Cardinal_function
same_as:
  wikidata: Q4352241
  wikipedia: Cardinal function
---

## Summary

A cardinal function in topology assigns a [[cardinal number|0EBACQFAZW]] to each topological space so that homeomorphic spaces get the same cardinal.
Standard examples are the weight, the density, the character and the cellularity.
Cardinal functions turn countability properties into numbers, so that, for example, [[second-countable|03PPNCM9GG]] means weight at most $\aleph_0$.
Many theorems of general topology are inequalities between cardinal functions.

## Explanations

### L1 human

A cardinal function measures the size of some feature of a space.
The weight counts how many open sets you need in a base, and the density counts how many points you need in a dense subset.
For the real line, both numbers are countable: open intervals with rational endpoints form a base, and the rationals are dense.
For an uncountable set with the discrete topology, both numbers equal the number of points.