ON STRONG FORM OF ARZELA CONVERGENCE

We define some new type of convergence of nets of functions which is formulated in terms of open covers. It preserves continuity and under some assumptions implies (or coincides with) the Arzela quasi-uniform convergence. Furthermore, the introduced strong convergence is used for characterization of compactness and regularity of a topological space.


INTRODUCTION
Let f, :X Y, n _> 1, be continuous functions and let f :X Y be a limit of the sequence {f, :n _> 1}. Which assumptions on the convergence guarantee the continuity of f? This question led to defining various types of convergences for nets of functions with values in metric or uniform spaces [1,2,3,4,5]. But initial notions in this problontinuity and pointwise convergence---are depending on topologies only and they can be considered even then if spaces are not uniformizable. This is the motivation of our paper. We define some new form of convergence formulated in terms of open covers; it preserves continuity and under some assumptions implies (or coincides with) the Arzela quasiuniform convergence. Let X, Y be topological spaces; the symbols F(X, Y) and C(X, Y) are ued to denote the class of all functions or all continuous functions from X to Y, respectively. For any set A its closure is denoted by CIA.
this net is pointwise convergent to f, and for each open cover A of Y and each j0 3' there exists a finite set J0 c 3' such that j _> J0 for j J0 and for each x 6 x there are j J0 and W A with {f(z), fs (x)} C W THEOREM 1. Let X be a topological space and Y a regular one. Ira net {f j J} C C(X, Y) is strongly convergent to a function f X -+ Y, then f 6 C(X, Y). PROOF The sequence {g, n > 1} strongly converges to g but not uniformly.
it is pointwise convergent to f, and for each V V and J0 5 J there exists a finite set J0 c J with j _> j0 for j 5 J0 and for each z X there is j J0 such that (f(z), f.(z)) V [1,2].
Let us remark that the quasi-uniform convergence is strictly connected with a uniformity (or metric) For instance let X Y (0,oo), da(x,y) x-Yl and (x,y) Ix 1; then dl and d2 are topologically equivalent metrics on Y. Putting f,(x) x + 1 f(x) x for z X, n > 1 we have the sequence {f," n > 1} which converges to f all-uniformly but it is not de-quasi-uniformly convergent to f.
PROPOSITION. Yet Y be a completely regular space. If a net {f'j J} C F(X,Y) is strongly convergent to a function f-X--, Y, then for each compatible uniformity Y on Y this net V-quasi-uniformly converges to f PROOF. Let [6].
It is easy to see that if V has the Lebesgue property, then for each X the strong convergence in F(X, Y) coincides with the V-quasi-uniform one.
The Lebesgue property is closely related to the paracompactness, namely we have the following THEOREM 2 [6]. Let   (b) X is an almost compact space; (c) for each regular space Y the strong convergence in C(X, Y) is equivalent to the pointwise one; then (a) (b) (c).
PROOF. We will show the implication (a) = (b). Let t. be an open cover of X and let (Us" s S} be the family of all finite sums of sets belonging to A.. We define a relation _< in S assuming s _< s iff Us C U; so (S, _< is a directed set. Now let us consider functions fs, f" X [0,1], E S, given by f(x) 1 for :v X and 1, if ClUs fs(z) 0, if z . X\CI It is easy to see that f is the pointwise limit of the net {f s S); thus, by the assumption this net strongly converges to f. Hence for the open cover g; { [0, ), (1/4, 43-), (, 1]) of [0, 1] and a fixed so E S there exists a finite set Sx C S with s _> so for $1 and for each z X some s e $1 can be taken such that both f(z) and fs(:r) are contained in the same set from G. This implies that for each z X there is s E S for which f(z) 1, which means X {CIUs s S} From the definition of the sets Us it follows that .A contains a finite family .A with X I,.J(Cl U U A }, so X is almost compact Now, we suppose that X is almost compact, Y is a regular space and {f j J) C C(X, Y) is a net of functions which pointwise converges to some f C(X, Y). Let