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Projections of zero-sets (and the fine uniformity on a product)

1969; American Mathematical Society; Volume: 140; Linguagem: Inglês

10.1090/s0002-9947-1969-0242114-2

1088-6850

Anthony W. Hager,

Functional Equations Stability Results

Introduction.We present here some results asserting that, under certain conditions on the pair of topological spaces (X, Y), the projection nx of Xx y onto X is "z-closed", i.e., carries zero-sets onto closed sets.These results are intended to contribute to the description of the fine uniformity on a product space, via the following.(Proof in §6, see also [N, 1.6].)1.1.The semi-uniform product X* Y of fine uniform spaces X and Y is fine iff trx is z-closed.(The terminology on uniform spaces follows [lx].We consider only completely regular HausdorfJ spaces.A zero-set is the set of zeros of a real-valued continuous function.)For comparison with our results, we state the following theorem, due to Isbell, using results and methods of Glicksberg, Frolik, and Onuchic.1.2 [lx, Chapter VII].The uniform product of two fine uniform spaces is fine iff either (a) for some cardinal n, one factor is discrete of power Sn, and the other is n-discrete, or, (b)fior some cardinal n, the product is pseudo-n-compact and m-discrete for allm<n.(Some of these terms are defined below.)Since the uniformity of the semi-uniform product is finer (larger) than that of the uniform product, each set of conditions in 1.2 is sufficient that rrx be z-closed.(In fact, see [N, 1.7].)We point out explicitly that we have not obtained a complete classification of circumstances under which nx is z-closed.See §5 for a discussion of this problem, and for remarks concerning the presumably simpler question of when -nx is closed.The latter, too, has not been completely answered although many results have been obtained (e.g., [HM], more extensively [N], [FF] and the references given there; here, 3.4 and §5.)