Pseudocompact Research Articles

A Separable Space with no Remote Points

In the model obtained by adding ${\omega _2}$ side-by-side Sacks reals to a model of ${\mathbf {CH}}$, there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size ${\omega _2}$ such that every uncountable subset has density ${\omega _1}$. Furthermore assuming the existence of a measurable cardinal $\kappa$ with ${2^\kappa } = {\kappa ^ + }$, a space $X$ is produced with no isolated points but with remote points in $\upsilon X - X$. It is also shown that a pseudocompact space does not have remote points.

A separable space with no remote points

In the model obtained by adding ω 2 {\omega _2} side-by-side Sacks reals to a model of C H {\mathbf {CH}} , there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size ω 2 {\omega _2} such that every uncountable subset has density ω 1 {\omega _1} . Furthermore assuming the existence of a measurable cardinal κ \kappa with 2 κ = κ + {2^\kappa } = {\kappa ^ + } , a space X X is produced with no isolated points but with remote points in υ X − X \upsilon X - X . It is also shown that a pseudocompact space does not have remote points.

Surfaces of codimension two with zero normal torsion carrying a conjugate coordinate net in Euclidean space

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New characterisations of pseudocompact spaces

In this paper, we give a new characterisation of pseudo-compact spaces, namely a space X is pseudocompact if and only if each σ-point finite open cover of X has a finite subfamily whose union is dense. As a corollary, we show that every pseudocompact σ-metacompact (or screenable) space is compact, which sharpens some known results.

The spaces Cp(X): decomposition into a countable union of bounded subspaces and completeness properties

A Tychonoff space X has to be finite if Cp(X) is σ-countably compact [23]. However, this is not true if only σ-pseudocompactness of Cp(X) is assumed. It is proved that Cp(X) is σ-pseudocompact iff X is pseudocompact and b-discrete. The technique developed yields an example showing that the theorem of Grothendieck [7] cannot be extended over the class of pseudocompact spaces. Some generalizations of the results of Lutzer and McCoy [9] are obtained. We establish also that ∏{Cp(Xt):tϵT} is a Baire space in case Cp(Xt) is Baire for each t ∈ T.

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G 0, G 1 such that G 0 × G 1 is not countably compact. In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenskiĭ, is this: In ZFC there are pseudocompact, homogeneous spaces X 0, X 1 such that X 0 × X 1 is not pseudocompact; if in addition MA is assumed, the spaces X i may be chosen countably compact. Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.

A class of extension properties that are not simply generated

Let P be a closed-hereditary topological property preserved by products. Call a space P -regular if it is homeomorphic to a subspace of a product of spaces with P . Suppose that each P -regular space possesses a P -regular compactification. It is well-known that each P -regular space X is densely embedded in a unique space γ scP X with P such that if f: X → Y is continuous and Y has P , then f extends continuously to γ scP X. Call P -pseudocompact if γ scP X is compact. Associated with P is another topological property P #, possessing all the properties hypothesized for P above, defined as follows: a P -regular space X has P # if each P -pseudocompact closed subspace of X is compact. It is known that the P -pseudocompact spaces coincide with the P #-pseudocompact spaces, and that P # is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if P is not the property of being compact and P -regular, then P # is not simply generated; in other words, there does not exist a space E such that the spaces with P # are precisely those spaces homeomorphic to closed subspaces of powers of E.

A wage-type example with a pseudocompact factor

The following example is constructed without any set-theoretic assumptions beyond ZFC: There exist a hereditarily separable hereditarily Lindelöf space X and a first-countable locally compact separable pseudocompact space Y such that dim X = dim Y = 0, while dim( X × Y)>0.

A characterization of pseudocompactness

A characterization of pseudocompactness

Pseudocompactness via graphs and projections

Pseudocompactness via graphs and projections

Approximation of continuous functions on pseudocompact spaces

If ${\mathcal {S}^ * }$ is the family of subrings of ${C^ * }(X)$ such that if $S \in {\mathcal {S}^ * }$, $S$ contains the constant functions and is closed under uniform convergence, then the following are equivalent for a space $(X,\mathcal {J})$. (a) $(X,\mathcal {J})$ is pseudocompact. (b) If $S \in {\mathcal {S}^ * }$ functionally separates points and zero sets, $S$ generates $(X,\mathcal {J})$. (c) If $S \in {\mathcal {S}^ * }$ functionally separates zero sets, $S = {C^ * }(X)$. (d) If $S \in {\mathcal {S}^ * }$ generates the zero sets on $(X,\mathcal {J}),S = {C^ * }(X)$. (e) If $f \in S \in {\mathcal {S}^ * }$ and $Z(f) = \phi$ then $1/f \in S$ (even when it is required that $S$ generate the topology). (f) If $f \in S \in \mathcal {S}$ then $\left | f \right | \in S$ (even when it is required that $S$ generate the topology).

Approximation of Continuous Functions on Pseudocompact Spaces

Approximation of Continuous Functions on Pseudocompact Spaces

Spaces with dense conditionally compact subsets

A subset S S of a topological space is said to be conditionally compact if every infinite subset of S S has a limit point in the space. If a space has a dense conditionally compact subset, it follows that it is pseudocompact, but the converse is not true. Examples are given of spaces that are pseudocompact, do not have dense conditionally compact subsets, but do have compactifications that are products of first countable spaces. For locally compact spaces, though, with such compactifications, the continuum hypothesis implies that pseudocompactness is equivalent to having a dense conditionally compact subset. A locally compact pseudocompact space without a dense conditionally compact subset is described.

Spaces with Dense Conditionally Compact Subsets

A subset $S$ of a topological space is said to be conditionally compact if every infinite subset of $S$ has a limit point in the space. If a space has a dense conditionally compact subset, it follows that it is pseudocompact, but the converse is not true. Examples are given of spaces that are pseudocompact, do not have dense conditionally compact subsets, but do have compactifications that are products of first countable spaces. For locally compact spaces, though, with such compactifications, the continuum hypothesis implies that pseudocompactness is equivalent to having a dense conditionally compact subset. A locally compact pseudocompact space without a dense conditionally compact subset is described.

A characterization of pseudocompactness

It is proved here that a completely regular Hausdorff space X is pseudocompact if and only if for any continuous function f from X to a pseudocompact space (or a compact space) Y, f*ϕ is z‐ultrafilter whenever ϕ is a z‐ultrafilter on X.

Pseudocompact metacompact spaces are compact

In 1950, Arens and Dugundji [1] defined metacompact spaces and showed (A) countably compact metacompact spaces are compact. It was known by then that paracompact spaces are normal and that normal pseudocompact spaces are countably compact. Since paracompact spaces are metacompact, (A) also showed (B) pseudocompact paracompact spaces are compact. The results (A) and (B) raise the following question of Aull and others [2]. as a common generalization: are pseudocompact metacompact spaces compact? This question was answered in the affirmative, first by Scott [3] and independently by Forster [4] and the author. The short proof given here also establishes an interesting property of Baire spaces. All spaces are assumed completely regular. A space is pseudocompact if every continuous real-valued function on it is bounded. A space is Baire if no open set is the union of countably many nowhere dense sets. A 7r-base for a space X is a family S of nonempty open subsets of X such that if G is a nonempty open subset of X, then some element of 3 is contained in G.

The Weight of a Pseudocompact (Homogeneous) Space Whose Cardinality Has Countable Cofinality

The Weight of a Pseudocompact (Homogeneous) Space Whose Cardinality Has Countable Cofinality

On nearly pseudocompact spaces

A completely regular space X is called nearly pseudocompact if υX− X is dense in β X− X, where β X is the Stone-Čech compactification of X and υ X is its Hewitt realcompactification. After characterizing nearly pseudocompact spaces in a variety of ways, we show that X is nearly pseudocompact if it has a dense locally compact pseudocompact subspace, or if no point of X has a closed realcompact neighborhood. Moreover, every nearly pseudocompact space X is the union of two regular closed subsets X 1, X 2 such that Int X 1 is locally compact, no points of X 2 has a closed realcompact neighborhood, and Int(X 1⌢X 2)=∅ . It follows that a product of two nearly pseudocompact spaces, one of which is locally compact, is also nearly pseudocompact.

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

Let X be an infinite pseudocompact space. We are interested in restrictions on κ = | X | \kappa = |X| and λ = w ( X ) \lambda = w(X) in addition to the obvious inequalities λ ⩽ 2 κ \lambda \leqslant {2^\kappa } and κ ⩽ 2 λ \kappa \leqslant {2^\lambda } and κ ⩾ 2 ω \kappa \geqslant {2^\omega } , valid for X without isolated points (in particular for homogeneous X). We show that if cf ( κ ) = ω {\text {cf}}(\kappa ) = \omega then λ ⩽ 2 > κ \lambda \leqslant {2^{ > \kappa }} , and even λ ⩽ 2 μ \lambda \leqslant {2^\mu } for some μ > κ \mu > \kappa 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.

A note on pseudocompact spaces

AbstractIn this note we give several new characterizations of arbitrary pseudocompact spaces, that is spaces characterized by the property that all continuous real-valued functions on the space are bounded.