# 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.

## 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 $X$ is termed a functionally Huasdorff space if, given any two points $x,y \in X$, there is a continuous function $f:X \to [0,1]$ such that $f(x) = 0$ and $f(y) = 1$.

## Proof

Suppose $X$ is a connected functionally Hausdorff space with at least two points. Say, $x \ne y \in X$ are two points. Then, by the functionally Hausdorff condition, there exists a function $f:X \to [0,1]$ such that $f(x) = 0$ and $f(y) = 1$.

Now, we claim that $f$ is surjective. Suppose not; suppose there exists $a \in (0,1)$ such that $f^{-1}(a)$ is empty. Then, $f^{-1}[0,a)$ and $f^{-1}(a,1]$ are disjoint open subsets whose union is $X$, and both are nonempty (because $f(x) = 0$ and $f(y) = 1$. This contradicts the assumption that $X$ is connected, hence $f$ must be surjective.

Thus, the cardinality of $X$ must be at least that of the continuum.