Products with closed projections. II

Noble, N.

doi:10.1090/S0002-9947-1971-0283749-X

Products with closed projections. II
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Trans. Amer. Math. Soc. 160 (1971), 169-183 Request permission

Abstract:

Conditions under which some or all of the projections on a product space will be closed or $z$-closed are studied, with emphasis on infinite products. These results are applied to characterize normal products up to countably many factors, to characterize closed product maps up to finitely many factors, and to give conditions under which products will be countably compact, Lindelöf, paracompact, $\mathfrak {m} - \mathfrak {n}$-compact, etc. Generalizations of these results to $\mathfrak {n}$-products and box products are also given. Our easily stated results include: All powers of a ${T_1}$ space $X$ are normal if and only if $X$ is compact Hausdorff, all powers of a nontrivial closed map $p$ are closed if and only if $p$ is proper, the product of countably many Lindelöf $P$-spaces is Lindelöf; and the product of countably many countably compact sequential spaces is countably compact sequential.

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16

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