# 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.

## Contents

## Statement

Any connected functionally Hausdorff space having at least two points is uncountable. In fact, its cardinality is at least equal to the cardinality of the continuum.

## Definitions used

### Connected space

`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 .

## Related facts

- 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.

## Proof

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.