Sandwich-type characterization of completely regular spaces
Keywords:Insertion, Sandwich theorem, Insertion theorem, Completely regular space, Lower semicontinuous, Upper semicontinuous, Compact.
AbstractAll the higher separation axioms in topology, except for complete regularity, are known to have sandwich-type characterizations. This note provides a characterization of complete regularity in terms of inserting a continuous real-valued function. The known fact that each continuous real valued function on a compact subset o fa Tychonoff space has a continuous extension to the whole space is obtained as a corollary.
R. L. Blair, Extension of Lebesgue sets and real valued functions, Czechoslovak Math. J. 31 (1981) 63–74.
G. Buskes and A. van Rooij, Topological Spaces. From Distance to Neighborhood, Springer, New York, 1997. http://dx.doi.org/10.1007/978-1-4612-0665-1
L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1976.
B. Hutton, Uniformities on fuzzy topological space, J. Math. Anal. Appl. 58 (1977) 559–571. http://dx.doi.org/10.1016/0022-247X(77)90192-5
P.T. Johnstone, Stone Spaces, Cambridge Univ. Press, Cambridge, 1982.
M. Katetov, On real-valued functions in topological spaces, Fund. Math. 38 (1951) 85–91; Correction: Fund. Math. 40 (1953) 203–205.
T. Kubiak, A strengthening of the Katetov-Tong insertion theorem, Comment. Math. Univ. Carolinae 34 (1993) 357–362.
E.P. Lane, Insertion of a continuous function, Top. Proc. 4 (1979) 463–478.
E. Michael, Continuous selections I, Ann. of Math. 63 (1956) 361–382. http://dx.doi.org/10.2307/1969615
H. Tong, Some characterizations of normal and perfectlynormal spaces, Duke J. Math. 19 (1952) 289–292. http://dx.doi.org/10.1215/S0012-7094-52-01928-5
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