About 122,000 results
|A topological space is called a-compact if it is the union of countably many |
compact sets, while a space is called Lindelof if every open cover has a
countable subcover. Clearly every compact space is a-compact and every a-
compact space ...
|5.3 Countable Compactness and Sequential Compactness DEFINITION 5.8. A |
set A is said to be countably compact if every infinite subset of A has a limit point
in A. Our next theorem shows that a compact set must always be countably ...
|Precompact Sets A set of a topological space is called a precompact set (|
countable- precompact set) or a relative compact set (a relative-countable-
compact set) if its closure is a compact set (a countable compact set). A
topological space is ...
|... 282Xq, 352Xk, 376Ym countable (set) 111F, 114G, 115G, §1A1, 226Yc, 4A1P, |
561A ideal of countable sets 556Xb; see ... 362Ye countably closed forcing 556R
, 556Xb, 5A3Q countably compact class of sets 413L, 413M, 4130, 413R, 413S, ...
|We say that a subset K C U is countably compact if every countable open |
covering of K has a finite subcovering. We say that K is relatively countably
compact if K is countably compact. It is clear that a compact set is countably
compact, and that ...
|(2) Let X be a separable locally compact metrizable space with compatible metric |
d. It suffices to show that every closed set is a Baire set. Now Lemma 2.76 implies
that X is a countable union of compact sets. Therefore each closed set is ...
|Definitions A topological space X is called countably compact provided that for |
every countable open cover U of X ... but not directly related to sequential
compactness, is ω-boundedness (every countable set is contained in a compact
|If the family is finite, the cover is called a finite cover; if it is countable or |
uncountable, the cover is accordingly called countable ... An analogous
argument proves that (0, l) is not a sequentially compact set in R. Definition—
Countably Compact ...
|The set R of real numbers is not compact, since R is contained in the union of all |
open intervals of unit length, but R is not contained in any finite union of such
intervals (see HEINE — Heine-Borcl theorem). A countably compact space is a ...
|(ii) ⇒ (i): If T is a quasi-Suslin map, each set T(α) is compact. (i) ⇒ (iii) is clear. (iii) |
⇒ (iv): Every Lindelöf space is realcompact and thus homeomorphic to a closed
subspace of a product of the real lines. (iv) ⇒ (v): Every relatively countably ...