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Uniform Separation and a Theorem of Katětov
Author(s) -
ITZKOWITZ GERALD
Publication year - 1995
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1995.tb55897.x
Subject(s) - uniform continuity , mathematics , uniform boundedness , subspace topology , lemma (botany) , uniform norm , pure mathematics , bounded function , norm (philosophy) , uniform limit theorem , logarithm , combinatorics , discrete mathematics , mathematical analysis , metric space , ecology , poaceae , political science , law , biology
In 1951, Katětov, in an article appearing in Fundamenta Mathematica, showed that each bounded uniformly continuous real‐valued function defined on a subspace ( Y Y ) of a uniform space ( X, ), can be extended to all of X so as to be uniformly continuous without increase in sup norm. The original proof made use of an intertwining lemma that tends to obscure the proof and its motivation. We define the notions of uniformly separated sets and functionally uniformly separated sets. It then follows from a classical and folklore theorem that two sets are uniformly separated iff they are functionally uniformly separated. Furthermore, it turns out that two sets are uniformly separated in Y iff they are uniformly separated in X . This allows us to apply an elementary Urysohn extension procedure argument to prove Katětov's theorem.