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On topological quotient maps preserved by pullbacks or products

  • B. J. Day (a1) and G. M. Kelly (a1)
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We are concerned with the category of topological spaces and continuous maps. A surjection f: XY 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: XY is every pullback of f a quotient map?

Question 2. For which continuous surjections f: XY is f × lz: X × ZY × 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 ×ZY.)

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

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References
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(1)Brown, R.Elements of modern topology. London (1968).
(2)Hájek, O.Notes on quotient maps. Comment. Math. Univ. Carolinae 7 (1966), 319323.
(3)Hájek, O.Correction to notes on quotient maps. Comment. Math. Univ. Carolinae 8 (1967), 171.
(4)Michael, E.Local compactness and cartesian products of quotient maps and k-spaces. Quart. J. Math. Oxford Ser., to appear.
(5)Michael, E.Bi-quotient maps and cartesian products of quotient maps. Ann. Inst. Henri Poincaré, Sect. B, to appear.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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