Improving Top: PrTop and PsTop.

*(English)*Zbl 0753.18003
Category theory at work, Proc. Workshop, Bremen/Ger. 1991, Res. Expo. Math. 18, 21-34 (1991).

[For the entire collection see Zbl 0732.00006.]

One of the most important properties of the category Top of all topological spaces are continuous maps is that it satisfies the initial condition and is well-fibred. A category of structured sets and structure-compatible maps satisfying these conditions is called a topological construct. Top does not possess, however, other desirable properties such as extensionality and Cartesian closedness. In this paper, the authors give methods of improving Top by its nice enlargements. Thus the categories PrTop of pretopological spaces and PsTop of pseudotopological spaces are constructed and the following properties are shown. (1) They are topological constructs. (2) Top is a bireflective subcategory of PrTop and PrTop is a bireflective subcategory of PsTop. (3) The embeddings in (2) are finally dense. (4) PrTop and PsTop are extensional. (5) PsTop is Cartesian closed. In other words, PrTop is an extensional topological hull of Top and PsTop is an extensional and Cartesian closed topological hull of Top. Though many results refer to other papers, the presentation of this paper is self- contained, except for proofs of theoretical background material on categorical topology, and is very clear.

One of the most important properties of the category Top of all topological spaces are continuous maps is that it satisfies the initial condition and is well-fibred. A category of structured sets and structure-compatible maps satisfying these conditions is called a topological construct. Top does not possess, however, other desirable properties such as extensionality and Cartesian closedness. In this paper, the authors give methods of improving Top by its nice enlargements. Thus the categories PrTop of pretopological spaces and PsTop of pseudotopological spaces are constructed and the following properties are shown. (1) They are topological constructs. (2) Top is a bireflective subcategory of PrTop and PrTop is a bireflective subcategory of PsTop. (3) The embeddings in (2) are finally dense. (4) PrTop and PsTop are extensional. (5) PsTop is Cartesian closed. In other words, PrTop is an extensional topological hull of Top and PsTop is an extensional and Cartesian closed topological hull of Top. Though many results refer to other papers, the presentation of this paper is self- contained, except for proofs of theoretical background material on categorical topology, and is very clear.

Reviewer: R.Nakagawa (Ibaraki)

##### MSC:

18B30 | Categories of topological spaces and continuous mappings (MSC2010) |

54A05 | Topological spaces and generalizations (closure spaces, etc.) |

54B30 | Categorical methods in general topology |