Nested sequences of local uniform spaces
Abstract: A locally uniform space is a pair of sets being a filter on such that , and . We shall say that a sequence is nested iff and . By a limit for a nested sequence , we shall mean any locally uniform space such that and . Our first task will be to consider when a nested sequence of locally uniform spaces has a limit; in order to do this, we shall introduce a weak generalization of pseudo-metric functions. We shall also show that, in contrast to locally uniform spaces, each nested sequence of uniform spaces has a limit.
With each locally uniform space one can associate a regular relative topology in the obvious fashion. E. Hewitt and J. Novak have constructed regular spaces of the type on which every real-valued continuous function is constant; we shall use our results about limits to give a relatively simple general construction for producing locally uniform spaces which have this type of relative topology. The construction may be done in such a way that the spaces produced have several pleasant topological properties.
-  Edwin Hewitt, On two problems of Urysohn, Ann. of Math. (2) 47 (1946), 503–509. MR 0017527, https://doi.org/10.2307/1969089
-  Josef Novák, Regular space, on which every continuous function is constant, Časopis Pěst. Mat. Fys. 73 (1948), 58–68 (Czech, with English summary). MR 0028576
-  J. R. Isbell, On finite-dimensional uniform spaces, Pacific J. Math. 9 (1959), 107–121. MR 0105669
-  James Williams, Locally uniform spaces, Trans. Amer. Math. Soc. 168 (1972), 435–469. MR 0296891, https://doi.org/10.1090/S0002-9947-1972-0296891-5
- E. Hewitt, On two problems of Urysohn, Ann. of Math. (2) 47 (1946), 503-509. MR 8, 165. MR 0017527 (8:165g)
- Josef Novak, Regular space on which every continuous function is constant, Časopis. Pěst. Mat. Fys. 73 (1948), 58-68. MR 10, 467. MR 0028576 (10:467h)
- J. R. Isbell, On finite dimensional uniform spaces, Pacific J. Math. 9 (1959), 107-121. MR 21 #4407. MR 0105669 (21:4407)
- James Williams, Locally uniform spaces, Trans. Amer. Math. Soc. 168 (1972), 435-469. MR 0296891 (45:5950)
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