A Note on Paracompact Spaces

نویسنده

  • ERNEST MICHAEL
چکیده

Let us quickly recall the definitions of the terms which are used in the statement of Theorem 1, and which will be used throughout this paper. Let X be a topological space. A collection <R of subsets of X is called open (resp. closed) if every element of "R. is open (resp. closed) in X. A covering of X is a collection of subsets of X whose union is X; observe that in this paper a covering need not be open. If 'R. is a covering of X, then by a refinement of 'R we mean a covering V of X such that every element of V is a subset of some element of 'R. A collection <R of subsets of X is locally finite if every x£X has a neighborhood which intersects only finitely many elements of <R. Finally, X is paracompact [3, p. 66] if it is Hausdorff, and if every open covering of X has an open, locally finite refinement. (Metric spaces and compact Hausdorff spaces are paracompact (cf. [il] and [3]), and every paracompact space is normal [3].) In §2 we prove Theorem 1, after first obtaining some preliminary lemmas, the first of which may have some independent interest. In §§2 and 3, we derive some of the consequences of Theorem 1 ; §2 deals with the relation of paracompactness to other topological properties, and §3 deals with subsets and cartesian products of paracompact spaces.

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تاریخ انتشار 2010