# Continuous Convergence on C(X) by Prof. Dr. Ernst Binz (auth.) By Prof. Dr. Ernst Binz (auth.)

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Extra info for Continuous Convergence on C(X)

Example text

A s s i g n i n g this limit, we obtain a r e a l - v a l u e d f u n c t i o n f. The con- can be shown as follows: e - O )~ s Cc(X) , p C X, ~((T For any point q s F to any point there are sets - T) x F) c U of for a given closed n e i g h b o r h o o d {t(q) - f(q) a filter G. ) 2 E Cc(X) , converging to f(q) G, e - e, converges to zero. Completeness that and (in In a d d i t i o n both are complete: In a convergence @ Ac we have is adherent to {t(q) I t s T, q s F} p C X T E e and and any filter F s @ with U, o 6 ~.

If finitely many members or X~M Mw s W convergent in belongs to for which X } filter ~, X~Mw W. we find belongs is a covering system of this system would cover X, then would have to contain the empty set. Let proves, X now be a compact convergence as in general topology, into a Hausdorff any continuous Moreover convergence real-valued Ac on Using ultrafilters that for any continuous space function C(X) space. Y the image of X f(X) induced by ~ : X the u n i f o r m topology. Xs, by theorem )Xs, Hence, from X is compact; thus is nothing but the supremum norm topology.

3 [ Bi 2 ] for the study of the ad- and another proof of corollary 18. c-embeddedness In this section we will turn our attention to those spaces which ix is a homeomorphism. [ Bi I ] , spaces [ Bi 2 ]. A characterization vergence spaces. c-embedded c-embedded of the c-embedded of c-embeddedness X topological is not too re- Cc(X) , where X spaces runs through all con- We close the section by a result which shows that convergence space We begin by collecting Cc(Y) all is Hausdorff X is determined by some basic properties for any Y, for spaces Moreover we will see that the class of all c-embedded is big enough to reproduce Since We call these spaces ensures us that the condition strictive.