Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-14T17:06:39.935Z Has data issue: false hasContentIssue false
  • English
  • Français

On topological quotient maps preserved by pullbacks or products

Published online by Cambridge University Press:  24 October 2008

B. J. Day  and
G. M. Kelly
B. J. Day
Affiliation:
University of New South Wales, Australia
G. M. Kelly
Affiliation:
University of New South Wales, Australia
Get access Rights & Permissions [Opens in a new window]

Extract

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−1G 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 × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)

Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 67 , Issue 3 , May 1970 , pp. 553 - 558
Copyright
Copyright © Cambridge Philosophical Society 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Brown, R.Elements of modern topology. London (1968).Google Scholar
(2)Hájek, O.Notes on quotient maps. Comment. Math. Univ. Carolinae 7 (1966), 319–323.Google Scholar
(3)Hájek, O.Correction to notes on quotient maps. Comment. Math. Univ. Carolinae 8 (1967), 171.Google Scholar
(4)Michael, E.Local compactness and cartesian products of quotient maps and k-spaces. Quart. J. Math. Oxford Ser., to appear.Google Scholar
(5)Michael, E.Bi-quotient maps and cartesian products of quotient maps. Ann. Inst. Henri Poincaré, Sect. B, to appear.Google Scholar