Acta Math. Hungar. 71 (4) (1996), 263-274. APPROXIMATE LIMITS OF m-COMPACT SPACES I. LON~AR (Vara~din) 1. Preliminaries The caxdinality of a set X will be denoted by IX I. The cofinality of a cardinal number m will be denoted by cf(m). A space X is called initially m-compact [16, p. 177] (where m is an infi- nite cardinal number) provided either of the following conditions holds. (i) for every filter-base ~r on X, if [~'l ~ m, then adx~" = N{C1F: F E ~'} ~ 9, or (ii) for every open cover/4, if 1511= m, then/4 has a finite cubcover. Ini- tially Ro-compact spaces are called initially compact or countably compact. It is obvious that each initially m-compact space is countably compact. A Tl-space X is strongly m-compact [16, p. 178] provided for every filter-base .T" on X, if I~'l ~ m, then there exists a compact subset K of X such that FMK ~ Oforall F E ~. Let/4 be any covering of a space X. For any subset Y of X we define st (Y,/4) = tA{UE/4: g O Y ~ 9}. Similarly, we define st//= {st (U,U): U E/4}. Inductively, for each positive integer n, st n 3/= st (st n-1 34), where st 134 = st/4. We say that a cover V is a star refinement of a cover/4 if the cover st 1; is a refinement of/4. An open cover )~ of a space X is normal [4, p. 379] if there exists a sequence }4;1, )4;2,... of open covers of the space X such that }4;1 = }4; and )4;i+1 is a star refinement of )IV/ for i = 1, 2, .... If/4, V E Cov (X) (i.e., are normal covqrs of X) and V refines/4, we write Y-~ ld. If f,g:Y ~ X are ~4-near mappings, i.e. if for any y E Y there ex- ists U E/4 with f(y), g(y) E U, then we write (f, g) -~/4. If X is a subspace of Y, and if/4 is a cover of Y, ~hen.,by the trace of 14 on X we mean the cover {U N X: UE/4}. The trace of/4 o~ X is denoted by//IX. If f: X -. Y is a continuous mapping and if/4 is a normal (respectively, locally finite) cover of Y, then f-1(/4) = {f-l(U): Ue/4} is a normal (respectively, lo- cally finite) cover of X [1, p. 13, Proposition 1.21]. If X _C y and if/4 is a normal (respectively, locally finite) cover of Y, then/4[X']'s a normal (re- spectively, locally finite) cover of X [1, p. 13 Proposition 1.22]. Suppose the n-tuple {/41,... ,/In} is a finite sequence of covers of X, then by the intersec- tion cover An=lUi we mean the family {U1 M... M Un: Ui E b/i, i = 1,..., n}. 0236-5294/96/$5.00 (~) 1996 Akad~ntiai KindS, Budapest