A nonpseudocompact product space whose finite subproducts are pseudocompact

Comfort, W. W.

doi:10.1007/bf01362285

Cited by 26 publications

(16 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[3]. Since (from Corollary 9.12 of [12]) every infinite compact subset of ßN\N has cardinality 2C, each compact subset of A u A is finite.…”

Section: Corollarymentioning

confidence: 99%

On the Hewitt Realcompactification of a Product Space

Comfort¹

1968

Transactions of the American Mathematical Society

Self Cite

View full text Add to dashboard Cite

One of the themes of Edwin Hewitt's fundamental and stimulating work [16] is that the g-spaces (now called realcompact spaces) introduced there, although they are not in general compact, enjoy many attributes similar to those possessed by compact spaces; and that the canonical realcompactification vX associated with a given completely regular Hausdorff space X bears much the same relation to the ring C(X) of real-valued continuous functions on X as does the Stone-Cech compactification ß^to the ring C*(X) of bounded elements of C(X). Much of the Gillman-Jerison textbook [12] may be considered to be an amplification of this point. Over and over again the reader is treated to a "/?" theorem and then, some pages later, to its "v" analogue.One of the most elegant "jS" theorems whose "v" analogue remains unproved and even unstated is the following, given by Irving Glicksberg in [13] and reproved later by another method in [11] by Zdenëk Frolik: For infinite spaces X and Y, the relation ß(Xx F)=JSXx^Fholds if and only if Zx Fis pseudocompact. The present paper is an outgrowth of the author's unsuccessful attempt to characterize those pairs of spaces (X, Y) for which v(Xx Y) = vXxvY. It is shown (Theorem 2.4) that, barring the existence of measurable cardinals, the relation holds whenever y is a ¿-space and uXis locally compact; and more generally (Theorem 4.5) that the relation holds whenever the ¿-space Y and the locally compact space vX admit no compact subsets of measurable cardinal. The question arises naturally as to when it will occur that vX is locally compact. Our best result in this direction, a part of Theorem 4.8, is, again, not definitive: X is locally pseudocompact if and only if there is a locally compact space Y for which X<^ F<= VX.Some of the questions treated here are susceptible to attack when thrown into the uniform space context. See Onuchic [18] and, more intensively, Isbell [17, especially Chapter 8].I am indebted to Tony Hager for several references, and for allowing me to announce here that, in addition to the results of [15], he has achieved certain new theorems on the relation v(Xx Y) = vXxvY, which being "uniform" and not "topological" have negligible overlap with those of this paper.

show abstract

“…[3]. Since (from Corollary 9.12 of [12]) every infinite compact subset of ßN\N has cardinality 2C, each compact subset of A u A is finite.…”

Section: Corollarymentioning

confidence: 99%

On the Hewitt Realcompactification of a Product Space

Comfort¹

1968

Transactions of the American Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…Becausê~ is productive, &* Π Jf = 0. (This space X is similar to spaces constructed in [2]; all are subspaces of βN.) (3) Let tc be an uncountable regular cardinal.…”

Section: (S T) Est{f{s T) and ) V(s T ) E S X T }mentioning

confidence: 99%

Cartesian-closed coreflective subcategories of Tychonoff spaces

Tashjian¹

1979

Pacific J. Math.

View full text Add to dashboard Cite

“…Terasaka's example (see [8, p. 135]) shows that if X is a countably compact completely regular space, then XxX need not be pseudocompact. Comfort's example in [4] shows that if {X(h) \ neN} are completely regular spaces, then n {X(n) \ne N} is not necessarily pseudocompact even if F] {X(n) | n e B) is pseudocompact for every finite (nonempty) subset B of N. On the other hand, Glicksberg [9] and Bagley, Connell, and McKnight [1] have proved that under certain supplementary hypotheses the product of a collection of pseudocompact completely regular spaces is pseudocompact.…”

Section: Proof (I) Each F(n)mentioning

confidence: 99%

“…We first observe that the following equations hold for any neN and Therefore,/((a, b) n *)e/(U W«)}) = U {/(!>))}<= U {Z(»»<=0. 4. Pseudocompact product spaces.…”

Section: Proof (I) Each F(n)mentioning

confidence: 99%

“…By definition, a pseudocompact space is a topological space on which every continuous real valued function is bounded. Pseudocompact spaces which are completely regular have often been studied [1], [4], [5], [7]- [10], [12], and [15], and some results have been obtained that apply to noncompletely regular pseudocompact spaces [1], [10], and [13]. In general, however, much more is known about completely regular pseudocompact spaces than is known about arbitrary pseudocompact spaces.…”

mentioning

confidence: 99%

See 1 more Smart Citation