Connected and T1 with at least two points implies infinite

Statement

A topological space that has at least two points, is a T1 space, and is a connected space, must be infinite.

Related facts

• Connected and regular with at least two points implies uncountable
• Connected and Urysohn with at least two points implies cardinality at least that of the continuum
• Connected and normal with at least two points implies cardinality at least that of the continuum
• Path-connected and T1 with at least two points implies uncountable

Tightness

We cannot conclude anything about the cardinality beyond the fact that it is infinite. This is because, for every infinite cardinal, there exists a connected $T_{1}$-space of that cardinality. The space is obtained by taking the cofinite topology on a set of that cardinality.

In particular, the countable space with cofinite topology is a countable space that is both connected and $T_{1}$.