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CORNELIU CX)NSTANTINESCU

g : = fo(p,ltis obvious that (p is increasing and (p{J)=F, Let ^efÇ. Then, obviously и (ßx{ß})e(5 and (p ( \J (B x {B})) = AnF.

Bed \ ВбЗг /

Bc=AnF Bc^AnF

Hence 9((5)i3g. Conversely let (i, A)eJ. Then

( p { { ( k , B)eJ I (к, ß) ^ (i, A)}) 3 Л n {;ceJ | к ^ i}e g

( jo ( ® ) cg , ф(©) = д. The property c) is obvious. The last assertion is obvious, f

PROPOSITION 1.4. Let (N, f) be a countably compact net on a topological space. If the sequence (/ («))„g n ^^^ ^ unique adherent point, then it converges to this adherent point.

Let X be the unique adherent point of the sequence (/(«))« g n ^^^ assume that the sequence does not converge to x. Then there exists a neighbourhood U of x and a subsequence {f{ni))kç^ of the sequence {f{n)\^^ such that /(wj^)^ t/ for any /:eN. Let ^^ be an adherent point of the sequence (/(«jk))jkgN- Then >' is an adherent point of the sequence (/(«))„eN different from x and this is a contradiction, t

U . Eberlem Spaces

A Hausdorff topological space will be called an Eberlein space if for any countably compact net (/, / ) on it and for any filter 5 on /, finer than the section filter of /, the adherence of the filter/(5) is non-empty. This is equivalent to the assertion that for any countably compact net (/, / ) on it and for any ultrafilter U on /, finer than the section filter of/, the ultrafilter / (U) is convergent.

An Eberlein closed set of a Hausdorff topological space Z is a subset Y oï X such that for any countably compact net (/, / ) on the topological subspace Y and for any ultrafilter и on /, finer than the section filter of / and such that / (?I) converges in X to a point xeZ, we have xsY. Any closed set of a Hausdorff topological space is Eberlein closed. The intersection of any family and, by Proposition 1.3, the union of any finite family of Eberlein closed sets are also Eberlein closed. If Y is an Eberlein closed set of X and if Z is a subspace of X then Yr\ Z is Eberlein closed in Z,

PROPOSITION 2.1. Let X be an Eberlein space and Y be a subset of X. Y is an Eberlein subspace if and only if it is Eberlein closed. In particular any closed sub- space of an Eberlein space is also an Eberlein space. If Y is Eberlein closed in X, then it is Eberlein closed in X for any finer topology.

The proof is obvious, t