compact space

(redirected from Sequentially compact)

compact space

[¦käm‚pakt ′spās]
(mathematics)
A topological space which is a compact set.
References in periodicals archive ?
Blaschke's Theorem is an extension of the classical Heine-Borel Theorem which states that every closed and bounded subset K [??] [R.sup.n] is sequentially compact. Many mathematicians have studied convergence of convex sets on different spaces ([1], [11], [12]):
Consequently B is weakly sequentially compact, so weakly compact by the Eberlein-Smulian theorem [3, pg.
Then X is sequentially compact non-compact and under (CH) it even has a compact resolution, see [21, Theorem 3.6].
We say S is forward sequentially compact if every sequence has a forward convergent subsequence with limit in S.
Let (X, d) be a forward sequentially compact space and T : X [right arrow] X backward contraction.
A I-sequentially continuous image of any sequentially compact subset of X is sequentially compact.
Note that K is a compact subset in a sequential space, K is sequentially compact. So there is a convergent subsequence {x} [union] {[x.sub.n[.sub.k]] : k [member of] N} of {x} [union] {[x.sub.n] : n [epsilon] N} that converges to x [member of] [f.sup.-1](y).