Connected and T1 with at least two points implies infinite

From Topospaces
Jump to: navigation, search


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

Related facts


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.