Some extension theorems for continuous functions

Most of this paper ( § § 2 6 ) will be devoted to answering this question and related questions. The related questions arise in connection with the concepts of absolute retract, absolute neighborhood retract, and absolute neighborhood extensor (in § 2 these are all defined and their interrelations and significance explained), and it is both convenient and natural to answer all the questions simultaneously. Assuming that the space Y of (2) is metrizable, we are able to answer these questions completely (thereby solving some heretofore unsolved problems of Arens [2, p. 19] and-Hu [18]) in Theorems 3.1 and 3.2 of § 3 ; § § 4 and 5 are devoted to proving these theorems. In § 6 we show by an example that things can go completely awry if Y is not assumed to be metrizable.