Connected and functionally Hausdorff with at least two points implies cardinality at least that of the continuum
This article gives the statement, and possibly proof, of a topological space satisfying certain conditions (usually, a combination of separation and connectedness conditions) is uncountable.
Further information: Connected space
Functionally Huasdorff space
Further information: functionally Hausdorff space
A topological space is termed a functionally Huasdorff space if, given any two points , there is a continuous function such that and .
- Connected and normal Hausdorff with at least two points implies cardinality at least that of the continuum: This follows because, by Urysohn's lemma, normal Hausdorff spaces are functionally Hausdorff.
- Connected and regular Hausdorff implies uncountable: Interestingly, this proof does not yield that the cardinality must be at least that of the continuum, only that it must be uncountable.
Suppose is a connected functionally Hausdorff space with at least two points. Say, are two points. Then, by the functionally Hausdorff condition, there exists a function such that and .
Now, we claim that is surjective. Suppose not; suppose there exists such that is empty. Then, and are disjoint open subsets whose union is , and both are nonempty (because and . This contradicts the assumption that is connected, hence must be surjective.
Thus, the cardinality of must be at least that of the continuum.