Abstract
Let Cp (X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp (X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp (X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp (X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement "every countable subset of X is closed".
Original language | English |
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Pages (from-to) | 1937-1942 |
Number of pages | 6 |
Journal | Topology and its Applications |
Volume | 156 |
Issue number | 11 |
DOIs | |
State | Published - Jun 15 2009 |
Keywords
- Domain representable space
- Function space
- Pointwise convergence topology
- Pseudo-radial space
- Pseudocomplete
- Strongly Choquet complete
- Transfinite sequence
- Weakly Choquet complete