In this section, we introduce the concept of compactness of topological spaces. After that, we discuss a relative version of this notion, that of a proper map. A technical section about Tychonoff’s theorem follows, which can be skipped at the first reading. We then turn to mapping spaces. As it is mainly the function spaces that are of interest in analysis, in topology, it is the spaces of continuous maps that provide extremely important examples of topological spaces.