Two examples on preimages of metric spaces

Noble, N.

doi:10.1090/S0002-9939-1972-0310840-8

Two examples on preimages of metric spaces
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Proc. Amer. Math. Soc. 36 (1972), 586-590 Request permission

Abstract:

Examples are given to show that the product of a compact ${T_1}$-space with a metric space need not be a k-space, and that the product of a countably compact Hausdorff space with a metric space need not be a quasi-k-space. These examples show that the separation assumptions cannot be omitted in the following two known results: Each Hausdorff perfect preimage of a k-space is a k-space. Each regular quasi-perfect preimage of a ${T_1}$ quasi-k-space is a quasi-k-space. Some related results, perhaps of independent interest, are also obtained.

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Products of quotient maps and spaces with weak topologies