F-RETRACTS, L-RETRACTS and WAE(N)

Periodica Mathematica Hungarica Vol. ~3 (3), (1991), pp. ~33-$36

F-RETRACTS, L-RETRACTS AND WAE(n)

F. S. MAHMOUD (Sohag)

Abstract

The notion of a weakly absolute extensor in the dimension n for the class of bicompacts (WAE(n)) is intruduced in [5]. In this paper, the notions of F-retraction and L-retraction are introduced and using these notions, some properties of WAE(n) are given.

1. Introduction

Here "compact Hausdorff space" means a bicompact space [5], "mapping" means a continuous function [3] and "the dimension n" means the covering dimension [2]. Let us recall that a subset A of a topological space X is functionally open (closed) [2] if there exists a mapping f : X ~ I such that A = f-l((o, 1])(A = f-l(0)). Let us recall that a bicompact X is a weakly absolute extensor in the dimension n (n = 0, 1, 2,..., ~) for the class of bicompacts [5], if for every bicompact B with dimB

REMARK. A retract of a space X may not be an F-retract of it.

Mathematics subject classification numbers, 1980/85. Primary 54C55, 54C99; Sec- ondary 54B17, 54C15, 54C20, 54C50. Key words and phrases. F-retract, L-retract, neighbourhood retract, weakly absolute retract, functionally closed set, weakly absolute extensor, neihbourhood retract.

Akaddmiai Kiadd, Budapest Kluwer Academic Publishers, Dordrecht 234 MAHMOUD: F-RETRACTS, L-RETRACTS AND WAE(N)

F-retraction and WAE(n)

The following proposition shows that a retract of a metrizable space is also F-retract.

PROPOSITION 1.1. If X is a metrizable space, A C X, and A is a retract of X, then A is an F-retrac~ of X.

PROOF. Let A be a retract of a metrizable space X. Since every retract of a Hausdorff space X is closed in X [3], if follows that A is closed. But, every closed subspace of a metrizable space X is functionally closed [2], so A is functionally closed and hence A is an F-retract.

PROPOSITION 1.2. Every topological product of a countable collection of WAE(n) belongs to WAE(n).

PROOF. Let {X~, : # E M} be any given countable collection of WAE(n) and let X denotes the topological product space of this collection. We will prove that X belongs to WAE(n). Let B by any bicompact with dimB < n and f : A , X any mapping defined on a closed subspace A of B. For each p E M, consider the composed mapping f~ = p~,f : A , X t, where p~ : X ..... ~ X~ denotes the natural projection. Since X~ E WAE(n), there exists a functionally closed subspace Z~ of B containing A and an extension g~ : Z~ ~ X~ of f~ over Z~. Since M is countable, the intersection Z of all Z~, # E M is a functionally closed subspace of B containing A [2]. Define a mapping g : Z ~ X by

g(z) = e z.

It is easy to verify that glA = f.

2. L-retraction and WAE(n)

It is clear that every F-retract of a space X is an L-retradt of X, but the converse may not be true as shown by the following example.

EXAMPLE 2.1. Let X = {a,b,c,d,e} and the topology rx on X be rx = {X, ~, {a, b, c), {d,e)). We define the mapping r : X ~ A = {a, b) by taking r(a) = r(c) = r(d) - r(e) = a and r(b) = b. It is clear that A is an L-retract of X but not an F-retract of X. Now, we give an example to show that L-retract of a space X need not be a neighbourhood retract of X. EXAMPLE 2.2. Let (R, U) be the real line with the usual topology and A - {0)U { wl:n = 1,2,3,... } C R. It is easy to verify that A is an L-retract of//but not a neighbourhood retract of R. Although, we obtain the following theorem: