Difference between revisions of "Peripherically-compact space"
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| − | A [[Topological space|topological space]] having a [[Base|base]] of open sets with compact boundaries. A completely-regular peripherically-compact space $X$ has compactifications with zero-dimensional remainder (in the sense of the dimension ind, cf. [[Compactification|Compactification]]; [[Remainder of a space|Remainder of a space]]; [[Dimension|Dimension]]). If each compact set $A\subset X$ is contained in another compact set $B\subset X$ for which in $X$ there is a countable fundamental system of neighbourhoods (e.g., when $X$ is metrizable), then the peripheral compactness of $X$ is equivalent to the existence of compactifications of $X$ with zero-dimensional remainder. | + | A [[Topological space|topological space]] having a [[Base|base]] of open sets with compact boundaries. A completely-regular peripherically-compact space $X$ has compactifications with zero-dimensional remainder (in the sense of the dimension ind, cf. [[Compactification|Compactification]]; [[Remainder of a space|Remainder of a space]]; [[Dimension|Dimension]]). If each compact set $A\subset X$ is contained in another compact set $B\subset X$ for which in $X$ there is a countable [[Defining system of neighbourhoods|fundamental system of neighbourhoods]] (e.g., when $X$ is metrizable), then the peripheral compactness of $X$ is equivalent to the existence of compactifications of $X$ with zero-dimensional remainder. |
====References==== | ====References==== | ||
Latest revision as of 06:21, 26 September 2017
A topological space having a base of open sets with compact boundaries. A completely-regular peripherically-compact space $X$ has compactifications with zero-dimensional remainder (in the sense of the dimension ind, cf. Compactification; Remainder of a space; Dimension). If each compact set $A\subset X$ is contained in another compact set $B\subset X$ for which in $X$ there is a countable fundamental system of neighbourhoods (e.g., when $X$ is metrizable), then the peripheral compactness of $X$ is equivalent to the existence of compactifications of $X$ with zero-dimensional remainder.
References
| [1] | H. Freudenthal, "Neuaufbau der Endentheorie" Ann. of Math. , 43 (1942) pp. 261–279 |
| [2] | H. Freudenthal, "Kompaktisierungen und Bikompaktisierungen" Indag. Math. , 13 : 2 (1951) pp. 184–192 |
| [3] | E.G. Sklyarenko, "Bicompact extensions of semibicompact spaces" Dokl. Akad. Nauk. SSSR , 120 : 6 (1958) pp. 1200–1203 (In Russian) |
Comments
These spaces are also called rim-compact spaces.
Spaces with the property that every compact subset is contained in a compact subset with a countable neighbourhood base are called spaces of countable type, see [a1].
References
| [a1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |
| [a2] | J.R. Isbell, "Uniform spaces" , Amer. Math. Soc. (1964) pp. Chapt. 7 |
Peripherically-compact space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peripherically-compact_space&oldid=31854