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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nested sequences of local uniform spaces


Author: James Williams
Journal: Trans. Amer. Math. Soc. 168 (1972), 471-481
MSC: Primary 54E15
DOI: https://doi.org/10.1090/S0002-9947-1972-0298617-8
MathSciNet review: 0298617
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Abstract: A locally uniform space is a pair of sets $ (X,\mathcal{V}),\mathcal{V}$ being a filter on $ X \times X$ such that $ \forall U \in \mathcal{V},\Delta (X) = \{ \langle x,x\rangle :x \in X\} \subse... ...,{U^{ - 1}} = \{ \langle y,x\rangle :\langle x,y\rangle \in U\} \in \mathcal{V}$, and $ \forall x \in X,\exists V \in \mathcal{V}:(V \circ V)[x] \subseteq U[x]$. We shall say that a sequence $ \{ ({X_n},{\mathcal{V}_n}):n \in \omega \} $ is nested iff $ \forall n \in \omega ,{X_n} \subseteq {X_{n + 1}}$ and $ {\mathcal{V}_{n + 1}}\vert{X_n} = {\mathcal{V}_n}$. By a limit for a nested sequence $ \{ ({X_n},{\mathcal{V}_n}):n \in \omega \} $, we shall mean any locally uniform space $ (X,\mathcal{V})$ such that $ X = \cup \{ {X_n}:n \in \omega \} $ and $ \forall n \in \omega ,\mathcal{V}\vert{X_n} = {\mathcal{V}_n}$. 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.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0298617-8
Article copyright: © Copyright 1972 American Mathematical Society