Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality $\mathfrak{c}$ if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and $σ$-countably tight compactum has cardinality $\mathfrak{c}$ remains open.
We also show that if an arbitrary product is $σ$-countably tight then all but finitely many of its factors must be countably tight.
We construct a normal countably tight $T_1$ space $X$ with $t(X_δ) >2^ω$. This is an answer to the question posed by Dow-Juhász-Soukup-Szentmiklóssy-Weiss. We also show that if the continuum is not so large, then the tightness of $G_δ$-modifications of countably tight spaces can be arbitrary large up to the least $ω_1$-strongly compact cardinal.
In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example of a regular separable scattered topological space which cannot be embedded into an Urysohn countably compact topological space but embeds into a Hausdorff countably compact space.
The weak tightness $wt(X)$ of a space $X$ was introduced in [11] with the property $wt(X)\leq t(X)$. We investigate several well-known results concerning $t(X)$ and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum $X$ such that $wt(X)=\aleph_0<t(X)$ under $2^{\aleph_0}=2^{\aleph_1}$. In particular, this demonstrates the celebrated Balogh's Theorem [5] does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if $X$ is a homogeneous com…
A topological space $X$ is said to be {\em $Y$-rigid} if any continuous map $f:X\rightarrow Y$ is constant. In this paper we construct a number of examples of regular countably compact $\mathbb R$-rigid spaces with additional properties like separability and first countability. This way we answer several questions of Tzannes, Banakh, Ravsky, as well as get a consistent example of $\mathbb R$-rigid Nyikos space. Also, we show that it is consistent with ZFC that for every cardinal $κ<\mathfrak c$ there exists a regular separable countably compact space $X$ which is $Y$-rigid with respect to any …
We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space X=(X,T) is countably compact iff it is countably subcompact relative to T. (iii) For every metric space X=(X,d), X is compact iff it is subcompact relative to T. We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable space is completely metrizable, (b) every countably subcompact metrizable space is subcompact, (c) every complete metrizable space is subcompact is relatively consist…
In this paper, some features of countably $α$-compact topological spaces are presented and proven. The connection between countably $α$% -compact, Tychonoff, and $α$-Hausdorff spaces is explained. The space is countably $α$-compact space iff every locally finite family of non-empty subsets of such space is finite is demonstrated. The countably $% α$-compact space with weight greater than or equal to $\aleph_0$ is the $α$-continuous image of a closed subspace of the cube $D^{\aleph_0}$ is discussed. The boundedness of $α$-continuous functions mapping $α$% -compact spaces to other spaces is clea…
We prove that a player $α$ has a winning strategy in the Banach--Mazur game on a space $X$ if and only if $X$ is F-Y countably $π$-domain representable. We show that Choquet complete spaces are F-Y countably domain representable. We give an example of a space, which is F-Y countably domain representable, but it is not F-Y $π$-domain representable.
Motivated by results of Juhász and van Mill in [13], we define the cardinal invariant $wt(X)$, the weak tightness of a topological space $X$, and show that $|X|\leq 2^{L(X)wt(X)ψ(X)}$ for any Hausdorff space $X$ (Theorem 2.8). As $wt(X)\leq t(X)$ for any space $X$, this generalizes the well-known cardinal inequality $|X|\leq 2^{L(X)t(X)ψ(X)}$ for Hausdorff spaces (Arhangel{\cprime}skiĭ~[1],Š}apirovskiĭ}~[18]) in a new direction. Theorem 2.8 is generalized further using covers by $G_κ$-sets, where $κ$ is a cardinal, to show that if $X$ is a power homogeneous compactum with a countable cover of …
The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is, in ZFC, a regular (or just Hausdorff) SP of cardinality greater than $\mathfrak c$. While this question is left open, we establish a number of non-trivial results that we list: 1. It is consistent with Martin's Axiom and $\mathfrak c =\aleph_2$ that there is a countably tight and compact SP of cardinality $2^{\mathfrak c}$. 2. If $κ$ is a measurable cardinal then in the forcing extension obtained by…
In this paper, we characterize stratifiable (or semi-stratifiable) spaces, and monotonically countably paracompact (or monotonically countably metacompact) spaces by expansions of locally upper bounded semi-continuous poset-valued maps. These extend earlier results for real-valued Locally bounded functions.
We introduce a new class of $\varkappa$-metrizable spaces, namely countably $\varkappa$-metrizable spaces. We show that the class of all $\varkappa$-metrizable spaces is a proper subclass of counably $\varkappa$-metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with $\varkappa$-metrizable spaces. We prove a generalization of a Chigogidze result that the Čech-Stone compactification of a pseudocompact countably $\varkappa$-metrizable space is $\varkappa$-metrizable.
Modifying the known definition of a Pytkeev network, we introduce a notion of Pytkeev$^*$ network and prove that a topological space has a countable Pytkeev network if and only if $X$ is countably tight and has a countable Pykeev$^*$ network. In the paper we establish some stability properties of the class of topological spaces with the strong Pytkeev$^*$-property.
We solve a long standing question due to Arhangel'skii by constructing a compact space which has a $G_δ$ cover with no continuum-sized ($G_δ$)-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every $G_δ$ cover has a $\mathfrak{c}$-sized subcollection with a $G_δ$-dense union and that in a Lindelöf space with a base of multiplicity continuum, every $G_δ$ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Vega's celebrated theorem on th…
We prove that the countable product of lines contains a Borel linear subspace $L\ne\mathbb R^ω$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes of ``large'' sets and Kuczma--Ger classes in the topological vector spaces $\mathbb R^n$ for $n\le ω$.
A product of compact normal spaces is normal; the product of a countably infinite collection of non-trivial spaces is normal if and only if it is countably paracompact and each of its finite sub-products is normal; if all powers of a space X are normal then X is compact: provided in each case that the spaces involved are T1. Here I examine the situation for infinite products not required to be T1 (or regular), extending or generalizing each of these facts. In addition, I prove some related results, give a number of examples, explore some alternative proofs, and close with some speculation rega…
An $ω_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $ω_1$-compact space is $σ$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $ω_1$-compact space of cardinality $\aleph_1$ is $σ$-countably compac…
A space $X$ is called {\it selectively pseudocompact} if for each sequence $(U_{n})_{n\in \mathbb{N}}$ of pairwise disjoint nonempty open subsets of $X$ there is a sequence $(x_{n})_{n\in \mathbb{N}}$ of points in $X$ such that $cl_X(\{x_n : n < ω\}) \setminus \big(\bigcup_{n < ω}U_n \big) \neq \emptyset$ and $x_{n}\in U_{n}$, for each $n < ω$. Countably compact space spaces are selectively pseudocompact and every selectively pseudocompact space is pseudocompact. We show, under the assumption of $CH$, that for every positive integer $k > 2$ there exists a topological group whose $k$-th power i…
In \cite{Chaber}, Chaber has proved that countably compact spaces with a quasi $G_{δ}$-diagonal are compact. We prove that initially $κ$% -compact spaces with a quasi $G_{κ}$-diagonal are compact, for any infinite cardinal $κ.
We investigate the class of continuous images of non-commutative Valdivia compact spaces, in particular its subclass of weakly non-commutative Cor- son countably compact spaces. A key tool is the study of non-commutative Corson countably compact spaces and their stability. The results are the non- commutative version of results by O. Kalenda (2003). Moreover, we present a study of retractional skeletons on Aleksandrov duplicates of ordinal spaces.
Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We say that the generic countably infinite group is of property $P$ if $P$ is comeager in $G$. We prove that every group property with the Baire property is either meager or comeager. We show that there is a comeager elementary equivalence class in $G$ but every isomorphism class is meager. We prove that the generic group is algebraically closed, simple, not fin…
In this paper we study $n$-Hausdorff homogeneous and $n$-Urysohn homogeneous spaces. We give some upper bounds for the cardinality of these kind of spaces and give examples. Additionally we show that for every $n>2$, there is no $n$-Hausdorff 2-homogeneous space. Finally, for any $n$-Hausdorff space we construct an $n$-Hausdorff homogeneous extension which is the union of countably many $n$-H-closed spaces.
In this paper for each cardinal $κ$ we construct an infinite $κ$-bounded (and hence countably compact) regular space $R_κ$ such that for any $T_1$ space $Y$ of pseudo-character $\leqκ$, each continuous function $f:R_κ\rightarrow Y$ is constant. This result resolves two problems posted by Tzannes in Open Problems from Topology Proceedings and extends results of Ciesielski and Wojciechowski and Herrlich.
Given a Banach space we consider the $σ$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the cofinality. We determine their values for separable Banach spaces, and approximate them for nonseparable Banach spaces. The remaining questions reduce to deciding if the following can be proved in ZFC for every nonseparable Banach space $X$: (1) $X$ can be covered by $ω_1$-many of its hyperplanes; (2) All subsets of $X$ of cardinalities less than ${\rm cf}([{\rm dens…
Jayne and Rogers proved that every function from an analytic space into a separable metric space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_σ$ set under it is again $F_σ$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers theorem at finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals $α$ …
