Pointwise Products of Uniformly Continuous Functions
DOI:
https://doi.org/10.5644/SJM.01.1.10Keywords:
Completion of a metric space, finitely chainable, uniformly continuous function, uniformly continuous set, uniformly isolated, W–B spaceAbstract
The problem of characterizing the metric spaces on which the pointwise product of any two uniformly continuous real - valued functions is uniformly continuous is investigated. A sufficient condition is given; furthermore, the condition is shown to be necessary for certain types of metric spaces, which include those with no isolated point and all subspaces of Euclidean spaces. It is not known if the condition is always necessary.
2000 Mathematics Subject Classification. Primary: 54C10, 54C30; Secondary: 20M20
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References
M. Atsuji, Uniform continuity of continuous functions of metric spaces, Pac. J. Math., 8 (1958), 11–16.
E. S. Elyash, G. Laush and N. Levine, On the product of two uniformly continuous functions on the line, Amer. Math. Monthly, 67 (1960), 265–267.
T. Isiwata, On uniform continuity of C(X), Sugaku Kenkyu Roku of Tokyo Kyoiku Daigaku, 2 (1955), 36–45 (Japanese).
K. Kuratowski, Topology, Vol. I, Acad. Press, New York and London, 1966.
N. Levine, Uniformly continuous linear sets, Amer. Math. Monthly, 62 (1955), 579–580.
N. Levine and N. J. Saber, On the product of real valued uniformly continuous functions in metric spaces, Amer. Math. Monthly, 72 (1965), 20–28.
N. Levine and W. G. Saunders, Uniformly continuous sets in metric spaces, Amer. Math. Monthly, 67 (1960), 153–156.
S. B. Nadler, Jr. and T. West, A note on Lebesgue spaces , Topol. Proc., 6 (1981), 363–369.
J. Nagata, On the uniform topology of bicompactifications, J. Inst. Polytech. Osaka City Univ., 1 (1959), 28–38.
W. R. Parzynski and P. W. Zipse, Introduction to Mathematical Analysis, McGraw - Hill, Inc., New York, 1982.
