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On Pták's double-limit theorems

Published online by Cambridge University Press:  20 January 2009

N.J. Young
N.J. Young
Affiliation:
The University, Glasgow, W.2
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Consider uniform spaces X and Y and a separately uniformly continuous real-valued function f on X × Y. The following question arises in the theory of games: under what conditions can f be extended to a separately continuous function on × Ŷ, where , Ŷ are the completions of X and Y respectively? Firstly observe that such an extension is not always possible. If X = Y = (0, 1] with the usual uniform structure and f(x, y) = xy then f is separately uniformly continuous but has no separately continuous extension to × Ŷ = [0, 1]2 since such an extension would satisfy f(0, .) = 0 on Y and f(., 0) = 1 on X and so would necessarily have a discontinuity in one argument at the origin.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 17 , Issue 3 , June 1971 , pp. 193 - 200
Copyright
Copyright © Edinburgh Mathematical Society 1971

References

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