Unlike most other concepts, the notion of completeness is not a topological invariant. But it is closely related to some important topological properties. Our considerations here are limited to some useful theorems in topology which find frequent applications in analysis. We study complete spaces in Sect. 10.1, and treat the completion of incomplete metric spaces in Sect. 10.2. Baire spaces and Baire category theorem are discussed in Sect. 10.3.