Extensions of functions which preserve the continuity on the original domain

✍ Scribed by Camillo Costantini; Alberto Marcone

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 222 KB
• Volume: 103
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00165-5

✦ Synopsis

We say that a pair of topological spaces (X, Y ) is good if for every A ⊆ X and every continuous f : A → Y there exists f : X → Y which extends f and is continuous at every point of A. We use this notion to characterize several classes of topological spaces, as hereditarily normal spaces, hereditarily collectionwise normal spaces, Q-spaces, and completely metrizable spaces. We also show that if X is metrizable and Y is locally compact then (X, Y ) is good and we answer a question of Arhangel'skii's about weakly C-embedded subspaces. For separable metrizable spaces our classification of good pairs is almost complete, e.g., if X is uncountable Polish then (X, Y ) is good if and only if Y is Polish as well. We also show that if Y is Polish and X metrizable then f can be chosen to be of Baire class 1.