Countable Subspace Research Articles

A compact space K is Corson compact if and only if Cp(K) has a dense lc-scattered subspace

We establish that a compact space K is Corson compact if and only if it can be embedded in a topological group which is a W-space in the sense of Gruenhage. As a application of this result we show that K is Corson compact if and only if it embeds in Cp(X) for a Lindelöf scattered space X. We also prove that a zero-dimensional compact space K is Corson if and only if Cp(K,{0,1}) is a continuous image of a Lindelöf scattered space. Besides, a zero-dimensional countably compact space X is ω-monolithic if and only if Cp(X) has a dense functionally countable subspace.

Star countable subspaces in products of ordinals

In this paper, we discuss the relationships among weakly star countability, almost star countability, star countability and ω1-compactness of subspaces of products of two ordinals. We mainly give the following:(1)For two ordinals ρ and σ, we give a characterization when a subspace X⊆(ρ+1)×(σ+1) is κ-compact, where κ>ω is regular.(2)For a subspace X of ω12, X is almost star countable if and only if it is ω1-compact.(3)There exists a weakly star countable subspace X⊆ω12 such that X is not almost star countable.(4)There exists an almost star countable subspace X⊆(ω1+1)2 such that X is not star countable.(5)For subspaces A and B of an ordinal, A×B is weakly star countable if and only if it is ω1-compact.

Exponential separability is preserved by some products

We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma$-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $|Y|\leq 2^{\omega_1}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $|Y|\leq 2^{\mathfrak{c}} $ is scattered, then $X$ is exponentially separable. A Lindelöf $\Sigma$-space $X$ must be exponentially separable provided that every $Y\subset X$ with $|Y|\leq {\mathfrak{c}}$ is scattered. Under the Luzin axiom ($2^{\omega_1}>{\mathfrak{c}} $) we characterize weak exponential separability of $C_p(X,[0,1])$ for any metrizable space $X$. Our results solve several published open questions.

On the Extension of Functions from Countable Subspaces

On the Extension of Functions from Countable Subspaces

On extension of functions from countable subspaces

Рассматриваются три промежуточных класса пространств $\mathscr{R}_1\subset \mathscr{R}_2\subset \mathscr{R}_3$ между $F$-пространствами и $\beta\omega$-пространствами. Приводится характеризация $\mathscr{R}_1$- и $\mathscr{R}_3$-пространств в терминах продолжения функций, доказывается, что классы $\mathscr{R}_1$-, $\mathscr{R}_2$-, $\mathscr{R}_3$- и $\beta\omega$-пространств не сохраняются стоун-чеховскими компактификациями.

Some Properties of the Covariant Functor Set of Exponential Type

In this paper, it is shown that the sets of all non-empty subsets Set (x) of a topological space X with exponential topology is a covariant functor in the category of -topological spaces and their continuous mappings into itself. It is shown that the functor Set is a covariant functor in the category of topological spaces and continuous mappings into itself, a pseudometric in the space Set (x) is defined, and compact, connected, finite, and countable subspaces of Set (x) are distinguished. It also shows various kinds of connectivity, soft, locally soft, and n - soft mappings in Set (x). One interesting example is given for the TOPY category. It is proved that the functor Set maps open mappings to open, contractible and locally contractible spaces and into contractible and locally contractible spaces.

First-countable spaces every quotient of which is orderable

We prove the following Main Theorem: Every Hausdorff quotient image of a first-countable Hausdorff topological space X is a linearly ordered topological space (LOTS) if and only if X is a metrizable space which is the union of a discrete subspace and a compact countable subspace. As a corollary we characterize 1) σ-compact spaces, 2) locally compact spaces, 3) separable spaces every quotient image of which is a LOTS.We examine also several natural examples of non-first-countable Hausdorff topological spaces X such that every quotient image of X is a LOTS.

Forcing with a coherent Souslin tree and locally countable subspaces of countably tight compact spaces

We apply a method of Todorcevic in showing that PFA(S) implies that the coherent Souslin tree S forces various strengthenings of the assertion that every locally countable space of size ℵ1 that has a countably tight compactification must be σ-discrete.

Every monotonically normal Čech-complete space is subcompact

We prove that every monotonically normal Čech-complete space is subcompact. We also show that the complement of a countable subspace of a compact space is subcompact.

On the topology of free paratopological groups. II

Let FP(X) be the free paratopological group on a topological space X. For n∈N, denote by FPn(X) the subset of FP(X) consisting of all words of reduced length at most n, and by in the natural mapping from (X⊕X−1⊕{e})n to FPn(X). In this paper a neighbourhood base at the identity e in FP2(X) is found. A number of characterisations are then given of the circumstances under which the natural mapping i2:(X⊕Xd−1⊕{e})2→FP2(X) is a quotient mapping, where X is a T1 space and Xd−1 denotes the set X−1 equipped with the discrete topology. Further characterisations are given in the case where X is a transitive T1 space. Several specific spaces and classes of spaces are also examined. For example, i2 is a quotient mapping for every countable subspace of R, i2 is not a quotient mapping for any uncountable compact subspace of R, and it is undecidable in ZFC whether an uncountable subspace of R exists for which i2 is a quotient mapping.

Note on reflexivity of some spaces of continuous integer-valued functions

Given a topological group G we denote by G∧ the group of characters on G and reflexivity of G means that the natural map from G to G∧∧ is a topological isomorphism.We show that for any zero-dimensional realcompact k-space X and a discrete finitely generated abelian group A, the group AX of continuous maps from X to A with pointwise addition and compact–open topology is reflexive, and we construct a countable non-reflexive closed subgroup of ZX, where X is a countable subspace of the plane (this group embeds as a closed subgroup in many copies of the product (∑Z)c of continuum of the discrete Specker group). We show also that, for metrizable separable X, analyticity of (ZX)∧ is equivalent to complete metrizability of X.

On properties of h-homogeneous spaces with the Baire property

We describe how to assign an h-homogeneous space b ( X , k ) with a dense complete subspace and of weight k to any strongly zero-dimensional metric space X of weight ⩽ k. We investigate the properties of such spaces and obtain the conditions when b ( X 1 , k ) is homeomorphic to b ( X 2 , k ) . The h-homogeneous separable space T which is a union of B ( ω ) and a countable subspace was constructed by E. van Douwen. Similarly, the h-homogeneous separable space S which is a union of B ( ω ) and a σ-compact subspace was described by J. van Mill. These spaces are generalized for the non-separable case. We prove that if Ind X = 0 and X = G ∪ L , where G is an absolute G δ and L is of first category, then X ω is an h-homogeneous space. We consider certain cases when A × X ω is homeomorphic to X ω providing A ⊂ X ω .

Countably compact first countable subspaces of ordinals have the Sokolov property

A space X is Sokolov if for any sequence {Fn : n ∈ ℕ} where Fn is a closed subset of Xn for every n ∈ ℕ, there exists a continuous map f : X → X such that nw(f(X)) ≤ ω and fn(F n ) ⊂ Fn for all n ∈ ℕ. We prove that if X is a first countable countably compact subspace of an ordinal then X is a Sokolov space and CP(X) is a D-space; this answers a question of Buzyakova. Thus, for any first countable countably compact subspace X of an ordinal, the iterated function space Cp , 2n +1(X) is Lindelöf for any n ∈ ω Another consequence of the above results is the existence of a first countable Sokolov space of cardinality greater than c.

The structure of Valdivia compact lines

We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The conjecture asserted that a compact line is Valdivia compact if its weight does not exceed ℵ 1 , every point of uncountable character is isolated from one side and every closed first countable subspace is metrizable. It turns out that the last condition is not sufficient. On the other hand, we show that the conjecture is valid if the closure of the set of points of uncountable character is scattered. This improves an earlier result of the first author.

Projective versions of selection principles

All spaces are assumed to be Tychonoff. A space X is called projectively P (where P is a topological property) if every continuous second countable image of X is P . Characterizations of projectively Menger spaces X in terms of continuous mappings f : X → R ω , of Menger base property with respect to separable pseudometrics and a selection principle restricted to countable covers by cozero sets are given. If all finite powers of X are projectively Menger, then all countable subspaces of C p ( X ) have countable fan tightness. The class of projectively Menger spaces contains all Menger spaces as well as all σ-pseudocompact spaces, and all spaces of cardinality less than d . Projective versions of Hurewicz, Rothberger and other selection principles satisfy properties similar to the properties of projectively Menger spaces, as well as some specific properties. Thus, X is projectively Hurewicz iff C p ( X ) has the Monotonic Sequence Selection Property in the sense of Scheepers; βX is Rothberger iff X is pseudocompact and projectively Rothberger. Embeddability of the countable fan space V ω into C p ( X ) or C p ( X , 2 ) is characterized in terms of projective properties of X.

Variations of selective separability

A space X is selectively separable if for every sequence ( D n : n ∈ ω ) of dense subspaces of X one can select finite F n ⊂ D n so that ⋃ { F n : n ∈ ω } is dense in X. In this paper selective separability and variations of this property are considered in two special cases: C p spaces and dense countable subspaces in 2 κ .

Is Every Metric on the Cantor Set <em>σ</em>-Monotone?

The notions have applications in fractal geometry, see [2]. The following is proved in [1]. A metric space is monotone if and only if it is bi-Lipschitz equivalent to a 1-monotone space. A metric space with a dense monotone subspace is monotone. σ-monotone spaces have low topological dimension: If X is monotone and separable, then X (topologically) embeds into R and if X is σ-monotone, then its topological dimension is at most 1. But there are spaces with low dimension that are not σ-monotone: There exists a compact set X ⊂ R homeomorphic to [0, 1] that is not σ-monotone; in fact, each monotone subset of X is nowhere dense in X. It follows that X has a countable subspace that is not monotone, and a completely metrizable null-dimensional subspace that is not σ-monotone (recall that a topological space is null-dimensional if it has a base consisting of clopen sets). However, no example of a nulldimensional compact space that is not σ-monotone is known.

Unions of chains in dyadic compact spaces and topological groups

The following problem is considered: If a topological group G is the union of an increasing chain of subspaces and certain cardinal invariants of the subspaces are known, what can be said about G? We prove that if G is locally compact and every subspace in the chain has countable pseudocharacter or tightness, then G is metrizable. We also prove a similar assertion for σ-compact and totally bounded groups represented as the union of first countable subspaces, when the length of the chain is a regular cardinal greater than ω 1. Finally, we show that these results are not valid in general, not even for compact spaces.

On Countable Compactness and Sequential Compactness

If a countably compact 73 space X can be expressed as a union of less then c many first countable subspaces, then MA implies that X is sequentially compact. Also MA implies that every countably compact space of size < c is sequentially compact. However, there is a model of ZFC in which u,<c and there is a countably compact, separable 72 space of size u,, which is not sequentially compact. It is well known that every sequentially compact space is countably compact, but the reverse is false. Even compact spaces need not be sequentially compact. It is interesting to note that by adding some restrictions on the size of spaces, (countable) compactness of spaces is sometimes enough to guarantee sequential compactness. For instance, any compact space of size w, is sequentially compact (L) (also see (F, MS and W)). natural question is whether countably compact spaces of size < c are sequen- tially compact. The main result of this paper shows that it is undecidable in ZFC. w* means su\u, and for any P G u, P* = (C\su P)\P is a clopen subset of u*. Let A be a space with a dense subset = {ai: i < u}. Each x G X corresponds to a closed subset of «*; namely, Cx = H {{z < w: ai G U}*: U is a neighborhood of x}. Let t/* = {z: a, G U}*.

On countable compactness and sequential compactness

If a countably compact T 3 {T_3} space X X can be expressed as a union of less then c c many first countable subspaces, then MA implies that X X is sequentially compact. Also MA implies that every countably compact space of size &gt; c &gt; c is sequentially compact. However, there is a model of ZFC in which ω 1 &gt; c {\omega _1} &gt; c and there is a countably compact, separable T 2 {T_2} space of size ω 1 {\omega _1} , which is not sequentially compact.