# Connected and T1 with at least two points implies infinite

From Topospaces

## 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 -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 .