Products of M-Compact Spaces

Saks, Victor; Stephenson, R. M.

doi:10.2307/2037801

Cited by 18 publications

(16 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The following positive result is due to Stephenson and Vaughan [SV,Theorem 1.1]; it was proved earlier by Saks and Stephenson with the additional condition that all factors are regular [SS,Theorem 4.11].…”

Section: Introductionmentioning

confidence: 87%

“…The product of any number of initially K-compact spaces is initially K-compact provided k is singular and 2X < k for all X <k . G Saks and Stephenson ask whether [SS,p. 281], or not [SS,p.…”

Section: Introductionmentioning

confidence: 99%

“…G Saks and Stephenson ask whether [SS,p. 281], or not [SS,p. 287], initial /c-compactness is productive for (regular) k > co.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The product of two normal initially 𝜅-compact spaces

Douwen¹

1993

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We prove that it is independent from ZFC that for every cardinal k the following statements are equivalent:(a) K is singular; (b) initial K-compactness (defined above the introduction) is productive; (c) initial K-compactness is finitely productive; and (d) the product of two initially K-compact normal spaces is initially kcompact.In particular, MA implies that there are two countably compact normal spaces whose product is not countably compact.In our terminology Tx implies regular and normal, and k , X, and p denote infinite cardinals. Recall that a space is called initially K-compact if every open cover of cardinality at most k has a finite subcover, and that initial cocompactness is equivalent to countable compactness, for Tx -spaces. The meaning of initially < K-compact should be clear.

show abstract

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The product of two normal initially 𝜅-compact spaces

Douwen¹

1993

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Then X = U aGa is a dense countably compact subgroup of II with 1^1 = k. (This construction is taken from [SS,4.5], and goes back to [F, 2.9].) Next, assume k" ¥= k > 2U, but there is p < k (hence p < 2K) with X < 2».…”

Section: Consequently All Examples Below Have Weight Xmentioning

confidence: 99%

The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality

Douwen¹

1980

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let I be an infinite pseudocompact space. We are interested in restrictions on k = \X\ and X -w(X) in addition to the obvious inequalities A < 2" and k < 2\ and k > 2", valid for X without isolated points (in particular for homogeneous X). We show that if cf(ic) -u, then X < 2<", and even X < 2'' for some n < k if X is homogeneous. Under the Singular Cardinals Hypothesis (which is much weaker than the GCH), there are no further restrictions for X without isolated points.

show abstract

“…Turning to other topologies on C(X) we note that the compact-open topology arises in Theorem 1 only because we need the Stone-Weierstrass Theorem (that a subalgebra is dense if it contains the constants and separates points). When dealing with a subalgebra generated by a set of cardinality n which separates points, compactness can be weakened to weak-n-X0-compactness (each n-fold open cover has a finite subcollection whose union is dense-such spaces are studied, for example, in [4]) so if wwX=n, then in the set-open topology generated by all weakly-n-K0-compact subsets of X, ÔC(X)^n. In case wwX=H0, an even stronger statement holds: C(X) is separable in the set-open topology generated by all relatively pseudocompact subsets of X.…”

Section: Propositionmentioning

confidence: 99%

The Density Character of Function Spaces

Noble¹

1974

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

Abstract.For topologies between the pointwise topology and the compact-open topology, the density character of C{X) (and, for certain spaces Z, C(X, Z)) is described in terms of a cardinal invariant of X. The Hewitt-Pondiczery theorem on the density character of product spaces follows as a corollary.1. Description. Except in Corollary 2, all hypothesized spaces are assumed to be completely regular Hausdorff. The set of continuous functions from X to Z is denoted by C(X, Z) or, when Z=R, by C(X).The density character, ÔX, of a space X is the least cardinality of a dense subset of X and the weight, wX, of X is the least cardinality of an open basis of X. We define the weak weight, wwX, of X to be the least of the cardinals w Y for Y a continuous one-to-one image of X. Theorem 1. Let X be any infinite space and let C(X) have any topology between the pointwise topology and the compact-open topology. Then ÔC(X) = wwX.All proofs will be given in the next section. Our remaining results allow Theorem 1 to be applied to yield information about 6C(X, Z) for suitable paces Z.Lemma. Let C(X,Z) and C(X,Z*) both have either the topology of uniform convergence on members of some fixed cover of X or the set-open topology generated by such a cover. IfZ is a retract ofZ* then ÔC(X, Z)Ô C(X, Z*). Proposition.For any topologies between the pointwise topology and the compact-open topology:(i) For any space Z, ôZ^ôC(X, Z).(ii) IfZ contains a nondegenerate arc, then ôC(X)^ôC(X, Z). (iii) If for each finite subset F of X there exists a function f in C(X,Z) such that F and f(F) have the same cardinality, then ôC(X)^ôC(X, Z) • ÔC(Z).

show abstract

Products of M-Compact Spaces

Cited by 18 publications

References 0 publications

The product of two normal initially 𝜅-compact spaces

The product of two normal initially 𝜅-compact spaces

The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality

The Density Character of Function Spaces

Contact Info

Product

Resources

About