Products of straight spaces with compact spaces
Submitted: 2013-11-15
|Accepted: 2013-11-15
|Downloads
Keywords:
Straight spaces, Convergent sequence, Precompact, Uniformly continuous
Supporting agencies:
Abstract:
A metric space X is called straight if any continuous real-valued function which is uniformly continuous on each set of a finite cover of X by closed sets, is itself uniformly continuous. Let C be the convergent sequence {1/n : n ϵ N} with its limit 0 in the real line with the usual metric. In this paper, we show that for a straight space X, X × C is straight if and only if X × K is straight for any compact metric space K. Furthermore, we show that for a straight space X, if X × C is straight, then X is precompact. Note that the notion of straightness depends on the metric on X. Indeed, since the real line R with the usual metric is not precompact, R×C is not straight. On the other hand, we show that the product space of an open interval and C is straight.
References:
M. Atsuji, Uniform continuity of continuous functions on metric spaces, Pacific J. Math. 8 (1958), 11–16. http://dx.doi.org/10.2140/pjm.1958.8.11
M. Atsuji, Uniform continuity of continuous functions on metric spaces, Canad. J. Math. 13 (1961), 657–663. http://dx.doi.org/10.4153/CJM-1961-055-9
A. Berarducci, D. Dikranjan and J. Pelant, An additivity theorem for uniformly continuous functions, Topology Appl. 146-147 (2005), 339–352. http://dx.doi.org/10.1016/j.topol.2003.05.007
A. Berarducci, D. Dikranjan and J. Pelant, Local connectedness and extension of uniformly continuous functions, preprint.
A. Berarducci, D. Dikranjan and J. Pelant, Products of straight spaces, preprint.
