Pseudocompact Research Articles

τ-Pseudocompact Mappings

We consider the problem of extending the notion of τ-pseudocompactness from spaces to continuous mappings, obtain conditions under which the product of τ-pseudocompact mappings is τ-pseudocompact. Since any space X can be considered as a continuous mapping from X into a singleton, we obtain consequences of the theorems on multiplicativity of τ-pseudocompactness for spaces. Thus, we study the notion of τ-pseudocompact mapping and some its properties similar to those of a pseudocompact space as well as consequences of the obtained assertions for spaces.

Singular sets and maximal topologies

Spaces which are maximal with respect to a semi-regular property are characterised. Furthermore, a method to construct such topologies is given. Consequently, new characterisations of maximal pseudocompact spaces and of maximal Q.H.C. spaces are presented. Known characterisations of maximal connected spaces and of maximal feebly compact spaces are given alternative proofs.

Pseudocompact Mal'tsev spaces

A theorem due to Comfort and Ross asserts that the product of any family of pseudocompact topological groups is pseudocompact. We generalize this theorem to the case of Mal'tsev spaces. A Mal'tsev operation on a space X is a continuous function ƒ:X 3 → X satisfying the identity ƒ(x, y, y) = ƒ(y, y, x) = x for all x, y ϵ X. A topological space is Mal'tsev if it admits a Mal'tsev operation. We prove that every Mal'tsev operation on a pseudocompact space X can be extended to a Mal'tsev operation on βX. It follows that: 1. (1) if X is a pseudocompact Mal'tsev space, then βX is Dugundji; 2. (2) the product of any family of pseudocompact Mal'tsev spaces is pseudocompact.

Additivity of metrizability and related properties

A topological property P is called n-additive in nth power (or weakly n-additive) if a topological space X has P as soon as X n = ∪{ Y i : i ϵ n} where all Y i have P. If P is n-additive in nth power for all natural n ⩾ 1, we say that P is weakly finitely additive. The main question we deal with in this paper is whether metrizability is weakly finitely additive. It was proved by Tkachuk (1994) that it is so in the class of regular spaces with Souslin property. Metrizability was also proved by Tkachuk (1994) to be weakly finitely additive in the class of Hausdorff compact spaces. We generalize this last result, showing that metrizability is weakly finitely additive in the class of regular pseudocompact spaces. We also prove that if X n is a regular Lindelöf space then it is metrizable if represented as a union of its n metrizable subspaces. We show that there is an example of a Tychonoff nonmetrizable space X such that X n is a union of two metrizable subspaces for all n ⩾ 1. The method of constructing this example can be used to solve several problems stated by Tkachuk (1994).

On the finite-dimensionality of topological products

It is proved that there exist integers e( k, l) ⩾ − 1 for k, l = − 1, 0, 1, … such that Ind X × Y ⩽ e(Ind X, Ind Y) if the space X × Y is normal (and Hausdorff), Y is locally compact paracompact (in particular, compact) and Ind X < ∞, Ind Y < ∞ (therefore any normal product of two finitedimensional in the sense of Ind spaces, one of which is locally compact paracompact is finite-dimensional in the same sense). Analogous assertions hold for any strongly paracompact product, any normal product with one metrizable factor and any normal product of a pseudocompact space and a k-space. Also it is proved that a strongly paracompact or a z-embedded subspace of a finitedimensional in the sense of Ind normal space is finite-dimensional in the same sense.

On Cα- compact subsets

For an infinite cardinal α, we say that a subset B of a space X is C α - compact in X if for every continuous function f : X → R α , [ B] is a compact subset of R α. This concept slightly generalizes the notion of α-pseudocompactness introduced by J.F. Kennison: a space X is α-pseudocompact if X is C α - compact in itself. If α = ω, then we say C- compact instead of C ω - compact and ω-pseudocompactness agrees with pseudocompactness. We generalize Tamano's theorem on the pseudocompactness of a product of two spaces as follows: let A ⊆ X and B ⊆ Y be such that A is z- embedded in X. Then the following three conditions are equivalent: (1) A × B is C α - compact in X × Y; (2) A and B are C α - compact in X and Y, respectively, and the projection map π : X × Y → X is a z α - map with respect to A × B and A; and (3) A and B are Cω- compact in X and Y, respectively, and the projection map π : X × Y → X is a strongly z α- map with respect to A × B and A (the z α - maps are the strongly z α - maps are natural generalizations of the z- maps and the strongly z- maps, respectively). The degree of C α - compactness of a C- compact subset B of a space X is defined by: ϱ( B, X) = ∞ if B is compact, and if B is not compact, then ϱ( B, X) = supα: B is C α- compact in X . We estimate the degree of pseudocompactness of locally compact pseudocompact spaces, topological products and Σ-products. We also establish the relation between the pseudocompact degree and some other cardinal functions. In the context of uniform spaces, we show that if A is a bounded subset of a uniform space ( X, U), then A is C α - compact in X ̂ , where ( X ̂ , U ̂ ), is the completion of ( X, U) iff f( A) is a compact subset of R α from every uniformly continuous function from X into R α; we characterize the C α - compact subsets of topological groups; and we also prove that if G i : i I is a set of topological groups 15 and A i is a C α - compact subset of G α for all i I, then Π iI A i is a C α - compact subset of Π iI G i .

On embeddings of the countable fan space

We consider the possibility to embed the countable fan space into a countably compact or pseudocompact space satisfying certain properties (good separation, countable tightness, countable character at all points except one, etc.).

On (D,H)-analytic Sets and their Topological Applications

An abstract concept of (D, H)-analytic sets is introduced and applied to characterizations of normality, perfect normality and Oz-spaces, as well as to some problems related to the theory of compactifications. A metrization theorem for pseudocompact spaces is deduced.

Sums, products, and mappings of weakly pseudocompact spaces

Each open subspace of a weakly pseudocompact space is either weakly pseudocompact or locally compact Lindelöf. A topological sum is weakly pseudocompact if and only if 1. (1) each summand is either weakly pseudocompact or locally compact Lindelöf and 2. (2) the sum is either compact or not Lindelöf. If X is realcompact and the lattice of compactifications of X is a b-lattice or if X is a not Čech-complete G δ -diagonal space then X is not weakly pseudocompact. Weak pseudocompactness is neither an inverse invariant of perfect maps nor an invariant of open maps with compact fibers. There is a not pseudocompact space in which each zero set is weakly pseudocompact.

A little more on the product of two pseudocompact spaces

A little more on the product of two pseudocompact spaces

Free topological groups and inductive limits

It is shown that for every Tichonoff pseudocompact space X the following conditions are equivalent: (a) the free topological group F( X) is an inductive limit of the increasing sequence of its closed subspaces { F n ( X): n∈ N}; (b) X n is normal and countably compact for every n∈ N. Conditions implying the homeomorphisms β( F n ( X))≈ F n ( βX, n∈ N, are found. Also, it is proved that if all finite powers of X are pseudocompact then the natural mappings i n of ( X⊕{ e}⊕ X −1) n to F n ( X) are z-closed for all n∈ N +.

Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups

It is proved that a Tychonoff pseudocompact group with continuous multiplication is a topological group. It is also proved that a Tychonoff countably compact group with separately continuous multiplication is a topological group. A continuous function defined on a product X × Y of pseudocompact spaces is extended to a separately continuous function defined on the product β( X) × β( Y) of their Stone-Čhech compactifications.

Uniform density and M−density for subrings of C(X)

This paper deals with the equivalence between u−density and m−density for the subrings of C(X). It was proved by Kurzweil that such equivalence holds for those subrings that are closed under bounded inversion. Here an example is given in C(N) of a u−dense subring that is not m−dense. It is deduced that the two types of density coincide only in the trivial case where these topologies are the same, that is, if and only if X is a pseudocompact space.

A connected pseudocompact space

We construct, under the continuum hypothesis, a first countable connected pseudocompact Tychonoff space without a dense relatively countably compact subset. We do this by blending an Ostaszewski-type construction with a Stephenson-type Ψ-space construction. We also show that any locally pseudocompact first countable regular space with a dense locally conditionally compact subset can be embedded in a pseudocompact first countable regular space. This sheds light on Stephenson's problem whether any locally pseudocompact first countable regular space can be embedded in a pseudocompact first countable regular space.

Gδ-open functionally bounded subsets in topological groups

This paper deals with functionally bounded sets of a topological group which are G δ -dense in their closure in the bilateral completion of the group. We prove that the functionally bounded sets with this property coincide, for topological groups, with a kind of functionally bounded sets introduced by Isiwata: the hyperbounded sets. We prove that when B is a hyperbounded set of a topological group G which has a G δ -open subset dense in B, then B has only one compatible uniformity, the one inherited from G. We also prove that for any functionally bounded G δ -open subset A of a topological group G, the space cl G A belongs to the Frolik's class B , i.e., the Cartesian product of cl G A by any pseudocompact space is a pseudocompact space. As a consequence we get that every compact topological group is the Stone-Čech compactification of any of its G δ -dense subsets.

Consistently, Must Perfect Pseudocompact Spaces Be Separable?

Consistently, Must Perfect Pseudocompact Spaces Be Separable?

Extending the discrete finite chain condition

Established results show that every zero dimensional, first countable, locally compact space X can be densely embedded in a pseudocompact space Z that is also zero dimensional, first countable and locally compact. the construction relies on a maximal almost disjoint collection of open sets. Because zero dimensionality is present, these sets can be chosen to be clopen and Z is regular as a consequence. Moreover, the set Z − X is closed discrete. Without zero dimensionality, it is still possible to use this mad family approach and an extension Y of X can be constructed where every discrete collection of open sets is finite. In general, the set Y − X need not be closed discrete. This is to be expected; after all, there exist spaces for which no first countable pseudocompactification can be produced by adding a closed discrete set.

Some cardinal generalizations of pseudocompactness

Some cardinal generalizations of pseudocompactness

Constructing regular 2-starcompact spaces that are not strongly 2-star-Lindelöf

It is shown that from any first-countable, zero-dimensional, locally compact, non-DFCC space X with the property that every nonempty open set has π-weight c one can construct a pseudocompact space Y + that is not strongly 2-star-Lindelöf. Such a space Y + would therefore be 1. (1) 2-starcompact, T 3 but not strongly 2-starcompact; 2. (2) 2-star-Lindelöf, T 3 but not strongly 2-star-Lindelöf. A space satisfying (1) has been constructed using CH and a Moore space with a σ-locally countable base satisfying (2) is known. The examples generated here are easier than these two spaces, require no set theory beyond ZFC and make the distinctions (1) and (2) simultaneously.

TOPOLOGICAL GROUPS AND DUGUNDJI COMPACTA

A compact space is called a Dugundji compactum if for every compact containing , there exists a linear extension operator which preserves nonnegativity and maps constants into constants. It is known that every compact group is a Dugundji compactum. In this paper we show that compacta connected in a natural way with topological groups enjoy the same property. For example, in each of the following cases, the compact space is a Dugundji compactum: 1) is a retract of an arbitrary topological group; 2) , where is a pseudocompact space on which some -bounded topological group acts transitively and continuously. Bibliography: 57 titles.