Back to Search
Start Over
Non-Extendability of Bounded Continuous Functions
- Source :
- Canadian Journal of Mathematics. 32:867-879
- Publication Year :
- 1980
- Publisher :
- Canadian Mathematical Society, 1980.
-
Abstract
- If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).
- Subjects :
- Pure mathematics
General Mathematics
Bounded function
Mathematics
Subjects
Details
- ISSN :
- 14964279 and 0008414X
- Volume :
- 32
- Database :
- OpenAIRE
- Journal :
- Canadian Journal of Mathematics
- Accession number :
- edsair.doi...........c823467702b42a4b715943b87b6ee1bb