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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 $\forall U \in \mathcal {V},\Delta (X) = \{ \langle x,x\rangle :x \in X\} \subseteq U,{U^{ - 1}} = \{ \langle y,x\rangle :\langle x,y\rangle \in U\} \in \mathcal {V}$, 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 ${\mathcal {V}_{n + 1}}|{X_n} = {\mathcal {V}_n}$. 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 $\forall n \in \omega ,\mathcal {V}|{X_n} = {\mathcal {V}_n}$. 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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Article copyright: © Copyright 1972 American Mathematical Society