Characterization of C(x) among its Subalgebras by R. B. Burckel

By R. B. Burckel

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AIK is continuously injected into v Clearly we may regard the Stone-Cech compactification of IN X K as a closed subset of IN X X. e. that (AIK)~ To show that A that if f n AllN X K separates the points of IN x X the points of the subset IN X K. E A and co , (A IK)~, and so surely then s~pllfnlKIIAIK is the extension of f to a continuous function on lNX X is the extension of C But this amounts to showing s~pllfnllA < (and this is obvious) , for if is separates the points of IN X K, it is then enough to show that because A IK then evidently < co (f n } fllNXK (fnIK}.

We show that P showing that if n p) u n C) (V is among the c fx} U C is Vn ' whence We are therefore finished ~f is void and we argue this by contradiction, P F¢ then this end we want a minimal P f has an isolated point. pre-image for P To and, of course, if e is a linearly ordered (by inclusion) set of compact subsets K of Zorn provides one: for each X yEp, f- l (y) n K F ¢ such that f{K) = P, then K E e for each so by compactness (and the linearity of the inclusion order on ~ f-l{y) KEC nK = f-l{y) n r: K.

Thus with ••• gn. belongs to B BIK =0 fj(X j ) = 1, gj(Y) x =C(K). x = O. and has value E Xo Let a,~ E ~ Then for any I x and functions gj(X j ) separates the points of We will show that each such that = 1, fj(X) fj E A a the at x and XO. has a compact neighborhood Consider the FX provided by the hypothesis and use the result of the first paragraph to fx E I find a function compact neighborhood of (Note that then g/fxlKx. ) Fx. h inside If Fx Let be a Ifxl > O. on which belongs to 9 Kx C(Kx )' so does be an extension of this function to f EA and so IIKx Ifx(X) I > O.

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