A study of multivalued functions

Carlos Jorge Do Rego Borges

Vol. 23 (1967), No. 3, 451–461

DOI: 10.2140/pjm.1967.23.451

Abstract

The primary purpose of this study is to determine which topological properties of a space are preserved by multivalued functions. Among other results, the following are proved:

(A) Let F : X → Y be a perfect map from X onto Y , with F(x)≠∅ for each x ∈ X, where X and Y are T₁-spaces whose diagonals are G_δ-sets. Then X is metrizable (stratifiable) if and only if Y is metrizable (stratifiable)-see Theorem 3.2.

(B) If F : X → Y is a multivalued Y -compact quotient map from a separable metrizable space X onto a regular first countable space Y with a G_δ-diagonal, then Y is separable metrizable (see Theorem 4.5).

(C) Every (usc-) lsc-function F from a closed subset of a stratifiable space X to a topological space Y admits a (usc-) lsc-extension to all of X (see Theorem 5.2).

Mathematical Subject Classification

Primary: 54.65

Milestones

Received: 21 September 1966

Revised: 30 March 1967

Published: 1 December 1967