Countable Pseudocharacter Research Articles

Weak countability axioms of coset spaces

In this paper, we investigate weak countability axioms of coset spaces. We first discuss biradial coset spaces. It is mainly shown that if H is a closed neutral subgroup of a topological group G, then (1) G/H is biradial ⇔ G/H is nested; (2) G/H is monotonically normal if G/H is nested; (3) G/H is metrizable ⇔ G/H is a biradial space with countable pseudocharacter. We also study coset spaces with certain point-countable covers. In particular, we prove that if H is a closed neutral subgroup of a topological group G, then (4) G/H is metrizable ⇔ G/H is determined by a point-finite cover consisting of bisequential spaces. In the end, we consider coset spaces with certain local countable networks. It is mainly shown that if H is a closed neutral subgroup of a topological group G, then (5) G/H is metrizable ⇔ G/H is an α4 and k-space with an ωω-base; (6) G/H is cosmic ⇔ G/H is a separable space with countable cn-character; (7) G/H is metrizable ⇔ G/H has countable cn-character provided G/H has the Baire property.

Topological groups with strong disconnectedness properties

Topological groups whose underlying spaces are basically disconnected, F-, or F′-spaces but not P-spaces are considered. It is proved, in particular, that the existence of a Lindelöf basically disconnected topological group which is not a P-space is equivalent to the existence of a Boolean basically disconnected Lindelöf group of countable pseudocharacter, that free and free Abelian topological groups of zero-dimensional non-P-spaces are never F′-spaces, and that the existence of a free Boolean F′-group which is not a P-space is equivalent to that of selective ultrafilters on ω.

Generalized metrizable properties and cardinal invariants in quotient spaces of quasitopological groups

In this paper, we study some generalized metrizable properties and cardinal invariants in quotient spaces of quasitopological groups. We mainly show that (1) If H is a neutral subgroup of a quasitopological group G such that G/H is Hausdorff, then G/H is weakly first-countable if and only if G/H is symmetrizable; (2) If H is a neutral subgroup of a quasitopological group G such that G/H is a Hausdorff countably compact space of countable pseudocharacter, then G/H is metrizable; (3) If H is a neutral subgroup of a quasitopological group G such that G/H is a q-space, then G/H is a β-space; (4) If H is a closed neutral subgroup of a quasitopological group G, then |G/H|≤2e(G/H)ψ(G/H).

Closure index of semitopological groups

In this article, we define a new cardinal function called the closure index of a semitopological group. The closure index ci(G) of a semitopological group G satisfying the T1-separation axiom lies between their character and pseudocharacter. For locally compact topological groups, these three cardinal functions coincide. We extend this result to the class of feathered topological groups and determine the value of the closure index for other classes of topological groups with compact-like properties. We also present several examples of topological groups with compact-like properties for which the three cardinal functions are pairwise distinct.We also find some classes of topological groups of countable pseudocharacter that have countable closure index.

Submetrizability of some classes of coset spaces

We mainly show that if H is a neutral subgroup of a paratopological group G such that G/H is a Hausdorff Lindelöf space with countable pseudocharacter. Then G/H can be condensed onto a separable metrizable space.We also prove that if H is a closed neutral subgroup of an ω-balanced paratopological group G such that G/H is a Hausdorff space with countable π-character. Then G/H is submetrizable.These two results answer two questions posed by Ling, He, and Lin in 2022.

On uniformly dense Lindelöf subspaces of function spaces

A set Y⊂Cp(X) is uniformly dense in Cp(X) if it is dense in the uniform topology on C(X). We construct a zero-dimensional σ-compact space X such that Cp(X) has a uniformly dense Lindelöf subspace while Cp(X) is not normal. This example answers several published open questions. Additionally, we obtain a version of a theorem of Reznichenko on ω-monolithicity, under MA+¬CH, of a compact space X if Cp(X) has a uniformly dense Lindelöf subspace. We also prove that if X is a dyadic compact then Cp(X) has a uniformly dense subspace of countable pseudocharacter.

On C-embedded subspaces of weakly Lindelöf spaces

We show that if the product space X=∏i∈IXi is weakly Lindelöf and a subspace Y of X fills all countable subproducts of X, then Y is weakly Lindelöf and C-embedded in X. In particular, if the product space X=∏i∈IXi is weakly Lindelöf and the factors Xi are Tychonoff and have countable pseudocharacter, then every dense C-embedded subspace Y of X is weakly Lindelöf. It is also proved that every dense C⁎-embedded subspace of an arbitrary product Π=∏i∈IXi of Eberlein compacta is pseudocompact (hence C-embedded in Π) and weakly Lindelöf.Finally we present an example of a Lindelöf topological group with a dense C-embedded subgroup that fails to be weakly Lindelöf.

On discrete reflexivity of Lindelöf degree and pseudocharacter

We establish that any Hausdorff discretely κ-Lindelöf space X must be κ-Lindelöf if t(X)≤κ. Besides, in κ-Lindelöf spaces whose tightness does not exceed κ, the property of having pseudocharacter ≤κ must be discretely reflexive. Under the hypothesis c<ωω we prove that countable pseudocharacter is discretely reflexive in Lindelöf Σ-spaces. If X is a Tychonoff space and ψ(D‾)≤κ for any discrete set D⊂Cp(X), then κ+ is a caliber of X, we have the inequality hd(X×X)≤2κ and ψ(Y‾)≤κ for any Y∈[Cp(X)]≤κ+; this implies that ψ(Cp(X))≤2κ. We also show that the property of having pseudocharacter ≤κ is discretely reflexive in Cp(X) whenever X is a compact space with t(X)≤κ.

For every space X, either CpCp(X) or CpCpCp(X) is ψ-separable

We establish that, for any Tychonoff space X, at least one of the function spaces CpCp(X) and CpCpCp(X) must have a dense subspace of countable pseudocharacter. If GCH holds, then at least one of the spaces Cp(X) and CpCp(X) has a dense subspace of countable pseudocharacter. We also prove that all iterated function spaces Cp,n(K) have a uniformly dense subspace of countable pseudocharacter whenever K is a Corson compact space. Our results solve several published open questions.

On some properties of perfect images of GO-spaces and generalized trees

Let PIGO be the class of perfect images of generalized ordered (GO) spaces. We show that if X∈PIGO, then the following properties are equivalent: X is paracompact; X is metacompact; X is meta-Lindelöf; X is subparacompact; X is submetacompact; X is weak δθ‾-refinable; X is weak θ‾-refinable; X is a D-space; X is an aD-space; X is transitively D; X is linearly D; No closed subspace of X is homeomorphic to a stationary subset of a regular uncountable cardinal; X is submeta-Lindelöf. We also show that if f:X→Y is a perfect mapping such that Y is a semi-stratifiable space and X is a T1-space with a Gδ⁎-diagonal, then X is a semi-stratifiable space.Let PIGT be the class of perfect images of generalized trees with Hausdorff generalized Sorgenfrey topologies. If X∈PIGT and X is a k-semistratifiable (stratifiable) space, then X is metrizable. If X∈PIGT, then X is first-countable if and only if X has a countable pseudocharacter. If f:L→X is a continuous irreducible mapping from a GO-space L (generalized tree L with a Hausdorff generalized Sorgenfrey topology) onto a space X with a regular Gδ-diagonal (Gδ⁎-diagonal), then the space L has a regular Gδ-diagonal (Gδ⁎-diagonal).

Domination by small sets versus density

Given an infinite cardinal κ, we prove that a regular space X must have density not exceeding κ if and only if every subset of X of power (2κ)+ is κ-dominated, i.e., contained in the closure of a set of cardinality κ. Thus, it is natural to study the spaces in which all subsets of cardinality not exceeding 2κ are κ-dominated; this property will be called exponential κ-domination. The spaces with exponential ω-domination first appeared in the paper [5] because they are weakly exponentially separable. We will prove that the class of spaces with exponential κ-domination is preserved by continuous images, open subspaces, Σ2κ-products and κ-unions. It is shown that the cardinal κ+ is a caliber of any space X with exponential κ-domination. A Čech-complete space features exponential κ-domination if and only if its density does not exceed κ. However, the density of countably compact spaces with exponential ω-domination has no upper bound. If a regular space X has a dense subspace Y with πχ(Y)≤2κ then d(X)≤κ if and only if X features exponential κ-domination. We conclude the paper with an example of a non-separable Tychonoff space of countable pseudocharacter with exponential ω-domination.

Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids

We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = ∏ i ∈ I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous homomorphism f : S → K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every continuous homomorphism f of S to K has a finite type, which means that f admits a continuous factorization through a finite subproduct of D. A similar conclusion is obtained for continuous homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results.

Baire category properties of some Baire type function spaces

We study the Baire type properties in the classes Bα(X,Y) of Baire-α functions and Bαst(X,Y) of stable Baire-α functions from a topological space X to a topological space Y, where α≥1 is a countable ordinal. Among others we prove the following results. If X is a normal space, then B1(X)=RX iff X is a Q-space. If X is a Tychonoff space of countable pseudocharacter, then: (i) for every ordinal α≥2, the spaces Bα(X) and Bαst(X) are Choquet and hence Baire, and (ii) B1(X) is a Choquet space iff X is a λ-space. Also we prove that for a Tychonoff space X the space B1(X) is meager if X is airy. We introduce a new class of almost K-analytic spaces which properly contains Čech-complete spaces and K-analytic spaces and show that for an almost K-analytic space X the following assertions are equivalent: (i) B1(X) is a Baire space, (ii) B1(X) is a Choquet space, and (iii) every compact subset of X is scattered. We show that the Baireness of the function space B1(X) is essentially a property of separable subspaces of X: if Y is a Polish space and X is a Y-Hausdorff space, then B1(X,Y) is Baire iff every countable subset of X is contained in a subspace Z⊆X such that the function space B1(Z,Y) is Baire and the restriction operator B1(X,Y)→B1(Z,Y), f↦f|Z, is surjective.

On the reflection of the countable chain condition

We study the question of when an uncountable ccc topological space $X$ contains a ccc subspace of size $\aleph_1$. We show that it does if $X$ is compact Hausdorff and more generally if $X$ is Hausdorff with $\mathrm{pct}(X) \leq \aleph_1$. For each regular cardinal $\kappa$, an example is constructed of a ccc Tychonoff space of size $\kappa$ and countable pseudocharacter but with no ccc subspace of size less than $\kappa$. We also give a ccc compact $T_1$ space of size $\kappa$ with no ccc subspace of size less than $\kappa$.

Feathered gyrogroups and gyrogroups with countable pseudocharacter

Topological gyrogroups, with a weaker algebraic structure than groups, have been investigated recently. In this paper, we prove that every feathered strongly topological gyrogroup is paracompact, which implies that every feathered strongly topological gyrogroup is a D-space and gives partial answers to two questions posed by A.V. Arhangel?ski?(2010) in [2]. Moreover, we prove that every locally compact NSS-gyrogroup is first-countable. Finally, we prove that each Lindel?f P-gyrogroup is Ra?kov complete.

More on the cardinality of a topological space

&lt;p&gt;In this paper we continue to investigate the impact that various separation axioms and covering properties have onto the cardinality of topological spaces. Many authors have been working in that field. To mention a few, let us refer to results by Arhangel’skii, Alas, Hajnal-Juhász, Bell-Gisburg-Woods, Dissanayake-Willard, Schröder and to the excellent survey by Hodel “Arhangel’skii’s Solution to Alexandroff’s problem: A survey”.&lt;/p&gt;&lt;p&gt;Here we provide improvements and analogues of some of the results obtained by the above authors in the settings of more general separation axioms and cardinal invariants related to them. We also provide partial answer to Arhangel’skii’s question concerning whether the continuum is an upper bound for the cardinality of a Hausdorff Lindelöf space having countable pseudo-character (i.e., points are Gδ). Shelah in 1978 was the first to give a consistent negative answer to Arhangel’skii’s question; in 1993 Gorelic established an improved result; and further results were obtained by Tall in 1995. The question of whether or not there is a consistent bound on the cardinality of Hausdorff Lindelöf spaces with countable pseudo-character is still open. In this paper we introduce the Hausdorff point separating weight Hpw(X), and prove that (1) |X| ≤ Hpsw(X)&lt;sup&gt;aL&lt;/sup&gt;&lt;sup&gt;c&lt;/sup&gt;&lt;sup&gt;(X)ψ(X)&lt;/sup&gt;, for Hausdorff spaces and (2) |X| ≤ Hpsw(X)&lt;sup&gt;ω&lt;/sup&gt;&lt;sup&gt;L&lt;/sup&gt;&lt;sup&gt;c&lt;/sup&gt;&lt;sup&gt;(X)ψ(X)&lt;/sup&gt;, where X is a Hausdorff space with a π-base consisting of compact sets with non-empty interior. In 1993 Schröder proved an analogue of Hajnal and Juhasz inequality |X| ≤ 2&lt;sup&gt;c(X)χ(X)&lt;/sup&gt; for Hausdorff spaces, for Urysohn spaces by considering weaker invariant - Urysohn cellularity Uc(X) instead of cellularity c(X). We introduce the n-Urysohn cellularity n-Uc(X) (where n≥2) and prove that the previous inequality is true in the class of n-Urysohn spaces replacing Uc(X) by the weaker n-Uc(X). We also show that |X| ≤ 2&lt;sup&gt;Uc(X)πχ(X)&lt;/sup&gt; if X is a power homogeneous Urysohn space.&lt;/p&gt;

If K is Gul'ko compact, then every iterated function space C,(K) has a uniformly dense subspace of countable pseudocharacter

We establish that C p ( X ) has a dense subspace of countable i -weight if and only if d ( C p ( X ) ) ⩽ c . It is also proved that the space C p ( K ) has a dense F σ -discrete subspace whenever K is Corson compact. If both spaces X and C p ( X ) are Lindelöf Σ, then for any natural n ⩾ 2 , the iterated function space C p , n ( X ) has a uniformly dense subspace of countable pseudocharacter. In the case of a Gul'ko compact space K , there is a uniformly dense subspace of countable pseudocharacter in C p , n ( K ) for any n ∈ N . This is a new result even for C p ( K ) given that K is Eberlein compact.

On large sequential groups

We construct, using $\diamondsuit $, an example of a sequential group $G$ such that the only countable sequential subgroups of $G$ are closed and discrete, and the only quotients of&nbsp;$G$ that have a countable pseudocharacter are countable and Fréc

On dense subspaces of countable pseudocharacter in function spaces

A set A⊆Cp(X) is uniformly dense in Cp(X) if, for any f∈Cp(X) and ε>0, there is g∈A such that |g(x)−f(x)|<ε for all x∈X. We prove that Cp(X) has a uniformly dense subspace that condenses onto a second countable space if and only if Cp(X) has cardinality of the continuum. This answers a question of Tkachuk published in 2003. It is also proved that Cp(X) has a dense Fσ-metrizable subspace if X is metrizable or Eberlein compact. If X is Corson compact or Cp(X) is a Lindelöf Σ-space, then it is possible to find a dense set Y⊆Cp(X) with countable pseudocharacter.

The countable type properties in free paratopological groups

A space X is of countable type (resp. subcountable type) if every compact subspace F of X is contained in a compact subspace K that is of countable character (resp. countable pseudocharacter) in X. In this paper, we mainly show that: (1) For a functionally Hausdorff space X, the free paratopological group FP(X)and the free abelian paratopological group AP(X) are of countable type if and only if X is discrete; (2) For a functionally Hausdorff space X, if the free abelian paratopological group AP(X) is of subcountable type then X has countable pseudocharacter. Moreover, we also show that, for an arbitrary Hausdorff μ-space X, if AP2(X) or FP2(X) is locally compact, then X is a topological sum of a compact space and a discrete space.