This paper is to present a categorical approach to basic concepts of general topology. The basic concepts are based on a finitely complete category \(\mathcal X\) with some additional conditions. The first additional conditions are axioms for factorization systems, namely we assume the existence of (mono, epi)-decompositions for every morphism and {epi} is orthogonal to {mono}. Some elementary facts on \({\mathcal X}\) are proved. In exercises, relations between concepts of category theory and classical general topology are exhibitited. As the next step, axioms for closed maps are introduced, namely, a class \({\mathcal F}\) of morphisms (which is called a class of \({\mathcal F}\)-closed maps) is given. \({\mathcal F}\) contains all isomorphims and is closed under composition, \({\mathcal F}\cap\{\text{mono}\}\) is stable under pullback, and \(g\cdot f\in{\mathcal F}\), \(f\in \{\text{epi}\}\Rightarrow g\in{\mathcal F}\). For example, a morphism \(d\) is said to be \({\mathcal F}\)-dense, if in any decomposition \(d= m\cdot h\) with an \({\mathcal F}\)-closed subobject \(m\), \(m\) is an isomorphism. \(d\) is \({\mathcal F}\)-dense \(\Leftrightarrow d\) is orthogonal to \(m\) for all \(m\in{\mathcal F}\cap\{\text{mono}\}\).
In this article, the most important concept is a proper map, which is defined in the following way: a morphism \(f: X\to Y\) is called \({\mathcal F}\)-proper if for every object \(Z\), every restriction of \(f\times 1_z: X\times Z\to Y\times Z\) is \({\mathcal F}\)-closed. Under this concept, a compact object is easily defined: an object \(X\) of \({\mathcal X}\) is \({\mathcal F}\)-compact if the morphism: \(X\to\) {the terminal} is \({\mathcal F}\)-closed. Under the same idea, an object \(X\) of \({\mathcal X}\) is \({\mathcal F}\)-separated (or \({\mathcal F}\)-Hausdorff), if the unique morphism \(!_X:X\to\) {the terminal} is \({\mathcal F}\)-separated. Therefore, we have the concept of \({\mathcal F}\)-Tychonoff, namely it is embeddable into an \({\mathcal F}\)-compact \({\mathcal F}\)-Hausdorff object. In such a way, a categorical setting of well-known classical general topology is obtained. Finally, the authors introduce an infinite product axiom and confirm the existence of Stone-Čech compactifications. There are many exercises which are very useful and helpful for general topologists. This article is an important survey to the successfully obtained categorical setting.
For the entire collection see [Zbl 1034.18001].