Topological continuous convergence. (English) Zbl 0566.54006
The continuous convergence structure is imposed on the function space LIM(X,Y) of all continuous functions from a convergence space X into a topological space Y. As might be expected from earlier studies when \(Y=R\), local compactness on X provides the key to making continuous convergence topological. If Y is regular, ordinary local compactness for X proves sufficient; a slightly stronger form of local compactness works for arbitrary Y. Under both sets of assumptions, continuous convergence on LIM(X,Y) coincides with the compact open topology. If X satisfies a strong regularity condition, the two versions of local compactness are equivalent, and in this case the converse result holds that X is locally compact whenever continuous convergence on LIM(X,Y) is topological.
Reviewer: D.C.Kent
MSC:
| 54C35 | Function spaces in general topology |
| 54A20 | Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) |
| 54B30 | Categorical methods in general topology |
| 54D45 | Local compactness, \(\sigma\)-compactness |
Keywords:
limit space; splitting limitierung; continuous convergence structure; local compactness; compact open topology; strong regularity conditionReferences:
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