Approximate limits of m-compact spaces

Ivan Lončar, Varaždin, Croatia

Keywords: Approximate inverse system and limit, initially m-compact spaces AMS Subject classification: 54B25, 54D30

Abstract: The conception of set convergence is used to study properties of approximate systems of mcompact spaces and their limits in the sence of S. Mardešić [6]. The main results of this paper are the following. (a) Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of non-empty initially |A|-compact Tychonoff spaces. If ∏ X_{a} is initially |A|-compact, then X = lim X is a non-empty initially |A|-compact Tychnoff space. If all spaces X_{a} are normal and if p_{a} : lim X → X_{a} are the projections, then the approximate map p = {p_{a} : lim X → X_{a}, a ∈ A} is an approximate resolution. (Theorems 2.4. and 2.20.). (b) Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of Tychonoff spaces of character ≤ |A| (of the covering weight cw(X_{a}) ≤ |A|). Let r = {r_{a} : X → X_{a}, a ∈ A} be an approximate resolution of |A|-compact Tychonoff space X. Then r is a limit of X (Theorem 2.24.).