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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  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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