On the Cartesian Product of Two Compact Spaces

Novák, Jan

doi:10.4064/fm-40-1-106-112

Cited by 86 publications

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“…For a = ^0, the class ^(fc$ 0 ) is not finitely productive: there exists a pseudocompact space X such that X x X is not pseudocompact. (This example is due to Novak in [16], and it also appears in Chapter 9 of [6].) Therefore, if co (^ (^0) The Supporting Institutions listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its content or policies.…”

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.

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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.

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“…It is not true that a product of pseudo-compact spaces must be pseudo-compact. Indeed Novak [6] has recently given an example of a pair of countably compact completely regular spaces whose product is not even pseudo-compact (8).…”

Section: Theoremmentioning

confidence: 99%

Stone-Čech compactifications of products

Glicksberg¹

1959

Trans. Amer. Math. Soc.

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“…In 1953 Novak [4] published an example to show that countable compactness is not preserved under products. Novak's example consists of taking two countably compact subspaces Ax and A2 of the Stone-Cech compactification ßN of the natural numbers A such that Ax U A2 = ßN and Ax n A2 = A; the product Ax X A2 is not countably compact because it contains an infinite closed discrete space.…”

Section: Introductionmentioning

confidence: 99%

“…

Additional conditions are thus necessary on one of the countably compact spaces A or F to ensure countable compactness of the product A x F. Some of the additional properties on A which will guarantee this are: sequentially compact, first countable, sequential, k. These properties and some proofs have been discussed in a paper of S. Franklin [2].

…”

mentioning

confidence: 99%

Products of countably compact spaces

Rishel¹

1976

Proc. Amer. Math. Soc.

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Abstract.Extensions of sufficient conditions for the product of two countably compact spaces to be countably compact, plus a relevant example.In 1953 Novak [4] published an example to show that countable compactness is not preserved under products. Novak's example consists of taking two countably compact subspaces Ax and A2 of the Stone-Cech compactification ßN of the natural numbers A such that Ax U A2 = ßN and Ax n A2 = A; the product Ax X A2 is not countably compact because it contains an infinite closed discrete space.Additional conditions are thus necessary on one of the countably compact spaces A or F to ensure countable compactness of the product A x F. Some of the additional properties on A which will guarantee this are: sequentially compact, first countable, sequential, k. These properties and some proofs have been discussed in a paper of S. Franklin [2]. Other properties which generate countably compact products in this manner are paracompactness and metacompactness, since either of these conditions, when added to countable compactness of a factor, makes the factor compact, and the product of a compact space with a countably compact space is well known to be countably compact. In what follows, assume all spaces Hausdorff.A space which is a generalization of /:-space (hence, of first countable and sequential space) has proved fruitful in some product theorems. This space is called a weakly-^ space. Definition. A topological space A is weakly-/: iff a subset F of A is closed in X if F fi C is finite for every compact C in A.This notation was introduced in [5]. Weakly-A: spaces can be used to generalize the results mentioned above.Theorem. If X and Y are countably compact spaces such that X is weakly-k, then the product X X Y is countably compact.Proof. Consider the sequence {(xn,yn): n G A} in A X F. If the sequence {xn} is closed, we are done, because there is a subsequence of [xn] with a limit point in A. So assume xn # xm for all m ¥= « and that [xn] is not closed. By the weakly-Ä: condition a compact C exists in A such that card (C n {-*"})

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On the Cartesian Product of Two Compact Spaces

Cited by 86 publications

References 0 publications

Cartesian-closed coreflective subcategories of Tychonoff spaces

Cartesian-closed coreflective subcategories of Tychonoff spaces

Stone-Čech compactifications of products

Products of countably compact spaces

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