## Locally uniform spaces

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- by James Williams PDF
- Trans. Amer. Math. Soc.
**168**(1972), 435-469 Request permission

## Abstract:

The axioms for a*locally uniform space*$(X,\mathcal {U})$ may be obtained by localizing the last axiom for a uniform space to obtain $\forall x \in X,\forall U \in \mathcal {U},\exists V \in \mathcal {V}:(V \circ V)[x] \subseteq U[x]$. With each locally uniform space one may associate a regular topology, just as one associates a completely regular topology with each uniform space. The topologies of locally uniform spaces with nested bases may be characterized using Boolean algebras of regular open sets. As a special case, one has that locally uniform spaces with countable bases have pseudo-metrizable topologies. Several types of Cauchy filters may be defined for locally uniform spaces, and a major portion of the paper is devoted to a study and comparison of their properties. For each given type of Cauchy filter,

*complete*spaces are those in which every Cauchy filter converges; to

*complete*a space is to embed it as a dense subspace in a complete space. In discussing these concepts, it is convenient to make the mild restriction of considering only those locally uniform spaces $(X,\mathcal {V})$ in which each element of $\mathcal {V}$ is a neighborhood of the diagonal in $X \times X$ with respect to the relative topology; these spaces I have called NLU-spaces. With respect to the more general types of Cauchy filters, some NLU-spaces are not completable; this happens even though some completable NLU-spaces can still have topologies which are not completely regular. Examples illustrating these completeness situations and having various topological properties are obtained from a generalized construction. It is also shown that there is a largest class of Cauchy filters with respect to which each NLU-space has a completion that is also an NLU-space.

## References

- Manuel P. Berri and R. H. Sorgenfrey,
*Minimal regular spaces*, Proc. Amer. Math. Soc.**14**(1963), 454–458. MR**152978**, DOI 10.1090/S0002-9939-1963-0152978-9 - Eduard Čech,
*Topological spaces*, Publishing House of the Czechoslovak Academy of Sciences, Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney, 1966. Revised edition by Zdeněk Frolík and Miroslav Katětov; Scientific editor, Vlastimil Pták; Editor of the English translation, Charles O. Junge. MR**0211373** - L. W. Cohen,
*On imbedding a space in a complete space*, Duke Math. J.**5**(1939), no. 1, 174–183. MR**1546117**, DOI 10.1215/S0012-7094-39-00518-1 - A. S. Davis,
*Indexed systems of neighborhoods for general topological spaces*, Amer. Math. Monthly**68**(1961), 886–893. MR**214017**, DOI 10.2307/2311686 - A. H. Frink,
*Distance functions and the metrization problem*, Bull. Amer. Math. Soc.**43**(1937), no. 2, 133–142. MR**1563501**, DOI 10.1090/S0002-9904-1937-06509-8 - Paul R. Halmos,
*Lectures on Boolean algebras*, Van Nostrand Mathematical Studies, No. 1, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR**0167440** - Edwin Hewitt,
*On two problems of Urysohn*, Ann. of Math. (2)**47**(1946), 503–509. MR**17527**, DOI 10.2307/1969089 - J. R. Isbell,
*Uniform spaces*, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR**0170323**, DOI 10.1090/surv/012 - John L. Kelley,
*General topology*, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR**0070144** - Kiiti Morita,
*On the simple extension of a space with respect to a uniformity. I*, Proc. Japan Acad.**27**(1951), 65–72. MR**48782** - Jun-iti Nagata,
*On a necessary and sufficient condition of metrizability*, J. Inst. Polytech. Osaka City Univ. Ser. A**1**(1950), 93–100. MR**43448** - S. A. Naimpally,
*Separation axioms in quasi-uniform spaces*, Amer. Math. Monthly**74**(1967), 283–284. MR**210080**, DOI 10.2307/2316024 - V. W. Niemytzki,
*On the “third axiom of metric space”*, Trans. Amer. Math. Soc.**29**(1927), no. 3, 507–513. MR**1501402**, DOI 10.1090/S0002-9947-1927-1501402-2 - A. H. Stone,
*Paracompactness and product spaces*, Bull. Amer. Math. Soc.**54**(1948), 977–982. MR**26802**, DOI 10.1090/S0002-9904-1948-09118-2 - Jingoro Suzuki,
*On the metrization and the completion of a space with respect to a uniformity*, Proc. Japan Acad.**27**(1951), 219–223. MR**48785** - Elias Zakon,
*On uniform spaces with quasi-nested base*, Trans. Amer. Math. Soc.**133**(1968), 373–384. MR**227930**, DOI 10.1090/S0002-9947-1968-0227930-4

## Additional Information

- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**168**(1972), 435-469 - MSC: Primary 54E15
- DOI: https://doi.org/10.1090/S0002-9947-1972-0296891-5
- MathSciNet review: 0296891