The minimum uniform compactification of a metric space

R. Grant Woods

Fundamenta Mathematicae (1995)

  • Volume: 147, Issue: 1, page 39-59
  • ISSN: 0016-2736

Abstract

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It is shown that associated with each metric space (X,d) there is a compactification u d X of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of u d X are presented, and a detailed study of the structure of u d X is undertaken. This culminates in a topological characterization of the outgrowth u d ℝ n ∖ ℝ n , where ( ℝ n , d ) is Euclidean n-space with its usual metric.

How to cite

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Grant Woods, R.. "The minimum uniform compactification of a metric space." Fundamenta Mathematicae 147.1 (1995): 39-59. <http://eudml.org/doc/212073>.

@article{GrantWoods1995,
abstract = {It is shown that associated with each metric space (X,d) there is a compactification $u_dX$ of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of $u_dX$ are presented, and a detailed study of the structure of $u_dX$ is undertaken. This culminates in a topological characterization of the outgrowth $u_dℝ^n ∖ ℝ^n$, where $(ℝ^n,d)$ is Euclidean n-space with its usual metric.},
author = {Grant Woods, R.},
journal = {Fundamenta Mathematicae},
keywords = {proximity; remainder; Smirnov compactification},
language = {eng},
number = {1},
pages = {39-59},
title = {The minimum uniform compactification of a metric space},
url = {http://eudml.org/doc/212073},
volume = {147},
year = {1995},
}

TY - JOUR
AU - Grant Woods, R.
TI - The minimum uniform compactification of a metric space
JO - Fundamenta Mathematicae
PY - 1995
VL - 147
IS - 1
SP - 39
EP - 59
AB - It is shown that associated with each metric space (X,d) there is a compactification $u_dX$ of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of $u_dX$ are presented, and a detailed study of the structure of $u_dX$ is undertaken. This culminates in a topological characterization of the outgrowth $u_dℝ^n ∖ ℝ^n$, where $(ℝ^n,d)$ is Euclidean n-space with its usual metric.
LA - eng
KW - proximity; remainder; Smirnov compactification
UR - http://eudml.org/doc/212073
ER -

References

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  1. [A] M. Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958), 1-16. Zbl0082.16207
  2. [B] D. Bellamy, A non-metric indecomposable continuum, Duke Math. J. 38 (1971), 15-20. Zbl0219.54030
  3. [Bl] R. L. Blair, Spaces in which special sets are Z-embedded, Canad. J. Math. 28 (1976), 673-690. Zbl0359.54009
  4. [BSw] R. L. Blair and M. A. Swardson, Spaces with an Oz Stone-Čech compactification, Topology Appl. 36 (1990), 73-92. Zbl0721.54018
  5. [C] S. Carlson, Completely uniformizable proximity spaces, Topology Proc. 10 (1985), 237-250. Zbl0609.54022
  6. [Če] E. Čech, Topological Spaces, Wiley Interscience, Prague, 1966. 
  7. [vD] E. K. van Douwen, Remote points, Dissertationes Math. 188 (1981). 
  8. [D] J. Dugundji, Topology, Allyn and Bacon, 1965. 
  9. [E] R. Engelking, General Topology, Polish Scientific Publishers, Warszawa, 1977. 
  10. [GJ] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, 1960. Zbl0093.30001
  11. [HY] J. Hocking and G. Young, Topology, Addison-Wesley, Reading, 1961. 
  12. [K] M. Katětov, On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85-91. Zbl0045.25704
  13. [LR] R. Levy and M. D. Rice, Techniques and examples in U-embedding, Topology Appl. 22 (1986), 157-174. Zbl0603.54017
  14. [Ma] K. D. Magill, Jr., The lattice of compactifications of a locally compact space, Proc. London Math. Soc. (3) 18 (1968), 231-244. Zbl0161.42201
  15. [M] D. Mattson, Discrete subsets of proximity spaces, Canad. J. Math. 31 (1979), 225-230. Zbl0409.54032
  16. [NW] S. Naimpally and B. Warrack, Proximity Spaces, Cambridge Univ. Press, 1970. Zbl0206.24601
  17. [PW] J. R. Porter and R. G. Woods, Extensions and Absolutes of Hausdorff Spaces, Springer, 1987. 
  18. [R] M. C. Rayburn, Proximities, unpublished manuscript. Zbl0990.54023
  19. [Ra] J. Rainwater, Spaces whose finest uniformity is metric, Pacific J. Math. 9 (1959), 567-570. Zbl0088.38301
  20. [S] Yu. M. Smirnov, Mappings of systems of open sets, Mat. Sb. (73) 31 (1952), 152-166 (in Russian). Zbl0047.16102
  21. [Wa] R. C. Walker, The Stone-Čech Compactification, Springer, New York, 1974. Zbl0292.54001
  22. [Wi] S. Willard, General Topology, Addison-Wesley, Reading, 1970. 
  23. [Wo1] R. G. Woods, Certain properties of βX - X for σ-compact X, doctoral dissertation, McGill Univ., Montreal, 1969. 
  24. [Wo2] R. G. Woods, Coabsolutes of remainders of Stone-Čech compactifications, Pacific J. Math. 37 (1971), 545-560. Zbl0203.24802

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