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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Products with closed projections. II

Author: N. Noble
Journal: Trans. Amer. Math. Soc. 160 (1971), 169-183
MSC: Primary 54.25
MathSciNet review: 0283749
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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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Keywords: Closed projections, <IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$z$">-closed projections, normal products, paracompact products, Lindel&#246;f products, countably compact products, <!– MATH $\mathfrak {m} - \mathfrak {n}$ –> <IMG WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img6.gif" ALT="$\mathfrak {m} - \mathfrak {n}$">-compact products, <!– MATH $\mathfrak {n}$ –> <IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\mathfrak {n}$">-products, box products, products of closed maps
Article copyright: © Copyright 1971 American Mathematical Society