3. Remarks

• Any metric space is a Hausdorff space.

• A frequent object of study is compact Hausdorff spaces, which include the compact metric spaces such as the real unit interval.

• Just as for metric spaces, for a subset $S$ of a topological space $X$, we can relativize to $S$: The relative topology on $S$ consists of the intersections of $S$ with the open sets of $X$.

• Two topological spaces are said to be homeomorphic if they are in one-to-one correspondence by a correspondence that preserves open sets. Equivalently, there is a pair of inverse continuous functions between them.