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- W2017005594 abstract "A discussion of proofs of sufficient conditions for the product of two countably compact spaces to be countably compact. In 1953, J. Novak settled an old question of Cech by exhibiting two countably compact spaces 1 whose product fails to be countably compact [Nk]. He produced countably compact subsets N1 and N2 of AN whose intersection was precisely N. Thus the diagonal of N1 x N2 is infinite, closed and discrete, ruling out countable compactness. Fortunately, this not too pleasant state of affairs is easily remedied by imposing an additional restriction on one of the factors. If, in addition to both factors being countably compact, one is also compact, or sequentially compact (Mrowka [M]), or first-countable (Ryll-Nardzewski [R]), or sequential ( Franklin [F]) , or a k-space (Noble [Nb]) , then their product is countably compact. (These last results can be regarded as successive improvements since each first-countable space is sequential and each sequential space is a k-space.) It follows at once that N1 and N2 can have none of these properties and, in particular, are not k-spaces. Recently, Burke [ B, Example 1.101, apparently unaware of Noble's result, has reproved that N1 is not a k-space. His argument is as follows: The intersection A of N with a compact subset K of N 1 is homeomorphic to the diagonal of clN A x cl N2A, which is itself countably compact since the first factor is compact and second countably compact. But the diagonal is closed and discrete and, hence, A is finite. Thus N meets every compact subset of N1 in a finite set but is not itself closed. This same argument actually proves Noble's result. Suppose that D is a countable, closed, discrete subset of the product of two countably compact spaces C1 and C2. If D = I(x ., ydl, we may assume that the correspondence Received by the editors November 11, 1973. AMS (MOS) subject classifications (1970). Primary 54B10, 54D20, 54D50, 54D55; Secondary 54C10." @default.
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- W2017005594 title "On products of countably compact spaces" @default.
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