The Continuity of Functions on Cartesian Products
N. Noble · Transactions of the American Mathematical Society · 1970
Introduction. A function/issequentially continuous if the restriction off to each convergent sequence (including its limit) is continuous, and/is/c-continuous if its restriction to each compact subspace is continuous.If each sequentially continuous (resp.real valued sequentially continuous) function with domain X is continuous, A' is a sequential space (resp.jH-space); /V-spaces and /cB-spaces are defined analogously.In this paper we are concerned with determining conditions under which sequentially continuous or /c-continuous functions on a product space X= \~\asA Xa will be continuous.Concerning sequentially continuous functions, our result includes conditions necessary and sufficient that a product of first countable spaces be a sequential space or an sB-space.Concerning Ac-continuous functions, we show that if each Xa is either first countable or locally compact, then each kcontinuous function on X with regular range is continuous, and that products of locally pseudocompact /cB-spaces are /cB-spaces.We also consider sequentially continuous and /V-continuous group homomorphisms, and show, for instance, that the property "each /c-continuous homomorphism with T0 range is continuous" is preserved under arbitrary products.All of these results are given in §5.§1 presents three fairly general conditions which can be combined to force the continuity of functions on product spaces, and these conditions are studied in § §2, 3 and 4. As incidental results we answer negatively a pair of questions : " Is the Hewitt-Nachbin realcompactification of a Fréchet space a /c-space ?" and " If X and Y are normal /V-spaces, is the /c-extension of Xx Y completely regular?" raised by W. W. Comfort and E. A. Michael respectively ( §2); answer a question posed by Keisler and Tarski in [10] by showing that a certain condition on cardinals is equivalent to measurability ( §3); and prove an analogue of Tychonoff's Theorem by showing that k-compactness (introduced in [4]) is preserved under arbitrary products ( §4). Conditions forcing continuity.Recall that a subspace of X= \~[aeA Xa is called a H-subspace if it has the form {x e X: 8(x, y) is countable} for some fixed y in X,