# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Nested sequences of local uniform spacesHTML articles powered by AMS MathViewer

by James Williams
Trans. Amer. Math. Soc. 168 (1972), 471-481 Request permission

## 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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