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On topological quotient maps preserved by pullbacks or products

1970; Cambridge University Press; Volume: 67; Issue: 3 Linguagem: Inglês

10.1017/s0305004100045850

1469-8064

Brian J. Day, G. M. Kelly,

Advanced Topics in Algebra

We are concerned with the category of topological spaces and continuous maps. A surjection f : X → Y in this category is called a quotient map if G is open in Y whenever f −1 G is open in X . Our purpose is to answer the following three questions: Question 1. For which continuous surjections f : X → Y is every pullback of f a quotient map? Question 2. For which continuous surjections f : X → Y is f × l z : X × Z → Y × Z a quotient map for every topological space Z ? (These include all those f answering to Question 1, since f × l z is the pullback of f by the projection map Y × Z → Y .) Question 3. For which topological spaces Z is f × 1 Z : X × Z → Y × Z a qiptoent map for every quotient map f ?