Extensions of proximity functions
Author:
Don A. Mattson
Journal:
Proc. Amer. Math. Soc. 26 (1970), 347-351
MSC:
Primary 54.30
DOI:
https://doi.org/10.1090/S0002-9939-1970-0264631-5
MathSciNet review:
0264631
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Abstract | References | Similar Articles | Additional Information
Abstract: Let ${P^ \ast }(X)$ be the algebra of bounded, real-valued proximity functions on a proximity space $(X,\delta )$, where $X$ is a dense subspace of a topological space $T$. In this paper we obtain several conditions which are equivalent to the following property: every member of ${P^ \ast }(X)$ has a continuous extension to $T$. Examples concerning these results are included, one of which shows that this extension property is distinct from ${C^ \ast }$-embedding.
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Additional Information
Keywords:
Proximity space,
<IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$p$">-subspaces,
algebras of bounded realvalued proximity functions,
Smirnov compactification,
<!– MATH ${C^ \ast }$ –> <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${C^ \ast }$">-embedding,
round filters,
gauges,
continuous extension of functions,
Stone-Čech compactification
Article copyright:
© Copyright 1970
American Mathematical Society