No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Hereditarily monotonically Sokolov spaces have countable network weight
Abstract
We provide an example to show that the monotone Sokolov property is not necessarily preserved under compact continuous images. Furthermore, we prove that if X ² ∖Δ X is either monotonically Sokolov or monotonically retractable, then X must be cosmic; and that if X is either hereditarily monotonically Sokolov or hereditarily monotonically retractable, then X has a countable network. Moreover, we characterize the Lindelöf Σ-property in C p -spaces on Alexandrov doubles, and show that the LΣ(LΣ(⩽ω))-property is equivalent to the LΣ(⩽ω)-property for the class of C p -spaces and for the class of Gul'ko spaces. These results solve some problems published by Kalenda [7], Tkachuk [23], García-Ferreira and Rojas-Hernández [5], and Molina-Lara and Okunev [11].
We present a few results and problems related to spaces of continuous functions with the topology of pointwise convergence
and the classes of LΣ(≤ ω)-spaces; in particular, we prove that every Gul’ko compact space of cardinality less or equal to is an LΣ(≤ ω)-space.
Given compact spaces X and Y, if X is Eberlein compact and Cp,n(X) is homeomorphic to Cp,n(Y) for some natural n, then Y is also Eberlein compact; this result answers a question posed by Tkachuk. Assuming existence of a Souslin line, we give an example of a Corson compact space with a Lindelöf subspace that fails to be Lindelöf Σ; this gives a consistent answer to another question of Tkachuk. We establish that every Σs-product of K-analytic spaces is Lindelöf Σ and Cp(X) is a Lindelöf Σ-space for every Lindelöf Σ-space X contained in a Σs-product of real lines. We show that Cp(X) is Lindelöf for each Lindelöf Σ-space X contained in a Σ-product of real lines. We prove that Cp(X) has the Collins–Roscoe property for every dyadic compact space X and generalize a result of Tkachenko by showing, with a different method, that the inequality w(X)≤nw(X)Nag(X) holds for regular spaces.
This chapter provides an overview of LΣ(κ)-spaces. If κ be a cardinal (finite or infinite), then a space X is an LΣ (≤κ)-space if there is a second-countable space M and a compact-valued upper semi-continuous mapping p : M → X such that p (M) = X and w(p (x)) ≤ κ for every x ∈ X (w(·) denotes weight). A space X is an LΣ (κ)-space if it is an LΣ(≤)‑space and is not an LΣ(≤ λ)-space for any λ < κ. A space X is an LΣ (<κ )-space if there is a second-countable space M and a compact-valued upper semi-continuous mapping p: M → X such that p (M) = X and w( p (x)) < κ for every x ∈ X. For finite k the definition says that the images of points under p have at most κ points. The class LΣ(<ω) is the class of all images of second-countable spaces under finite-valued upper semicontinuous mappings. The definitions also admit natural reformulations in terms of networks modulo compact covers in the spirit of the seminal article of K. Nagami.
We show that any ∑s-product of at most c-many L∑(≤ ω)-spaces has the L∑(≤ ω)property. This result generalizes some known results about L∑(≤ ω)spaces. On the other hand, we prove that every ∑s-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ∑s-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe property by condensations, Topology Proc. 42 (2012), 1-15]. Besides, we prove that if X is a simple Lindelöf ∑-space, then Cp(X) has the Collins-Roscoe property.
In this article, we mainly study certain families of continuous retractions
(r-skeletons) having certain rich properties. By using monotonically
retractable spaces we solve a question posed by R. Z. Buzyakova in \cite{buz}
concerning the Alexandroff duplicate of a space. Certainly, it is shown that if
the space X has a full r-skeleton, then its Alexandroff duplicate also has
a full r-skeleton and, in a very similar way, it is proved that the
Alexandroff duplicate of a monotonically retractable space is monotonically
retractable. The notion of q-skeleton is introduced and it is shown that
every compact subspace of is Corson when X has a full q-skeleton.
The notion of strong r-skeleton is also introduced to answer a question
suggested by F. Casarrubias-Segura and R. Rojas-Hern\'andez in their paper
\cite{cas-rjs} by establishing that a space X is monotonically Sokolov iff it
is monotonically -monolithic and has a strong r-skeleton. The
techniques used here allow us to give a topological proof of a result of I.
Bandlow \cite{ban} who used elementary submodels and uniform spaces.
In this paper we deal with some classes of spaces defined by networks and retractions, in particular we prove: Any closed subspace in a Σ-product of cosmic spaces is monotonically stable. A space X is monotonically retractable if and only if it is monotonically ω-stable and has a full retractional skeleton. Any monotonically retractable and monotonically ω-monolithic space is monotonically Sokolov, and as a consequence, any monotonically Sokolov and monotonically ω-stable space is monotonically retractable. Any closed subspace of a countably compact monotonically retractable space X is a W-set in X. These results generalize some results obtained in [18], [6], [8] and [10].
We prove that if X is a Lindelöf ∑-space such that t(X) ≤ K and πΧ(X) ≤ K then X has a π-base of order at most κ. This generalizes a theorem of Shapirovsky on existence of a π-base of order ≤ κ in any compact space of tightness ≤ re. A famous theorem of Gul'ko says that if X is compact and CP(X) is Lindelöf S then X is Corson compact, i.e., there exists an embedding of X into a E-product of real lines. We establish a general fact about Lindelöf ∑-spaces which implies that if both spaces X and Cp(X) are Lindelöf ∑ then there is a rich family of retractions on the space X. As a consequence, for any Tychonoff space X, if Cp(X) is Lindelöf ∑ then X can be condensed into a ∑-product of real lines. This gives an essential strengthening of the Gul'ko's theorem. As an application of this result we show that if Cp(X) is a Lindelöf E-space and p(X) = ω then nw(X) ≤ 2ω.
In this article we introduce the notion of monotonically retractable space and we show that: (1) Cp(X) is Lindelöf and a D-space whenever X is monotonically retractable. (2) If X is monotonically retractable then C p,2n(X) is monotonically retractable for any n ∈ ω. (3) Any first countable countably compact subspace of an ordinal is monotonically retractable. (4) Every closed subspace of a Σ-product of cosmic spaces is monotonically retractable. As a consequence of these results we conclude that Cp,2n+1(X) is Lindelöf and has the D-property for any n ∈ ω, whenever X is a first countable countably compact subspace of an ordinal; this answers a question posed by Tkachuk in [15].
Given a cardinal κ say that X is an LΣ(< κ)- space (LΣ(≤ κ)-space) if X has a countable network F with respect to a cover C of X by compact subspaces of weight strictly less than κ (less than or equal to κ, respectively), i.e., given any C ∈ C, we have w(C) < κ (w(C) ≤ κ) and, for any U ∈ τ(X) with C ⊂ U, there exists F ∈ F such that C ⊂ F ⊂ U. These concepts were introduced and studied by Kubís, Okunev and Szeptycki. We show that if Cp(X) is a Lindelö f Σ-space and |X| ≤ c, then Cp(X) is an LΣ(≤ ω)-space. This answers two questions of Kubiś, Okunev and Szeptycki. We also prove that if X is a space and Cp(X) has the LΣ(< ω)-property, then X is cosmic, i.e., nw(X) ≤ ω. This answers (in a stronger form) a question of Okunev published in Open Problems in Topology II.
This chapter discusses cardinal functions. Cardinal functions extend important topological properties as countable base, separable, and first countable to higher cardinality. Cardinal functions allow one to formulate, generalize, and prove results of a particular type in a systematic and elegant manner. The cardinal functions also allow one to make precise quantitative comparisons between certain topological properties. Prerequisite for work in cardinal functions is knowledge of cardinal and ordinal numbers and transfinite constructions. The chapter also describes the cardinal functions on the two most important classes of abstract topological spaces, namely, compact spaces and metrizable spaces. It also describes cardinal functions that are used to obtain bounds on the cardinality of a space. An infinite cardinal that is not regular is said to be singular. Theorems from combinatorial set theory play an important role in cardinal functions.
We introduce the monotone Sokolov property and show that it is dual to monotone retractability in the sense that X is monotonically retractable if and only if Cp(X)Cp(X) is monotonically Sokolov. Besides, a space X is monotonically Sokolov if and only if Cp(X)Cp(X) is monotonically retractable. Monotone retractability and monotone Sokolov property are shown to be preserved by RR-quotient images and FσFσ-subspaces. Furthermore, every monotonically retractable space is Sokolov so it is collectionwise normal and has countable extent. We also establish that if X and Cp(X)Cp(X) are Lindelöf Σ-spaces then they are both monotonically retractable and have the monotone Sokolov property. An example is given of a space X such that Cp(X)Cp(X) has the Lindelöf Σ-property but neither X nor Cp(X)Cp(X) is monotonically retractable. We also establish that every Lindelöf Σ-space with a unique non-isolated point is monotonically retractable. On the other hand, each Lindelöf space with a unique non-isolated point is monotonically Sokolov.
A space X has the Collins–Roscoe property if we can assign, to each x∈Xx∈X, a family G(x)G(x) of subsets of X in such a way that for every set A⊂XA⊂X, the family ⋃{G(a):a∈A}⋃{G(a):a∈A} contains an external network of A¯. Every space with the Collins–Roscoe property is monotonically monolithic. We show that for any uncountable discrete space D, the space Cp(βD)Cp(βD) does not have the Collins–Roscoe property; since Cp(βD)Cp(βD) is monotonically monolithic, this proves that monotone monolithity does not imply the Collins–Roscoe property and provides an answer to two questions of Gruenhage. However, if X is a Lindelöf Σ-space with nw(X)⩽ω1nw(X)⩽ω1 then Cp(X)Cp(X) has the Collins–Roscoe property; this implies that Cp(X)Cp(X) is metalindelöf and constitutes a generalization of an analogous theorem of Dow, Junnila and Pelant proved for a compact space X. We also establish that if X and Cp(X)Cp(X) are Lindelöf Σ-spaces, then the iterated function space Cp,n(X)Cp,n(X) has the Collins–Roscoe property for every n∈ωn∈ω.
We prove that if Cp(X) is a Lindelöf Σ-space, then Cp,2n+1(X) is a Lindelöf Σ-space for every natural n. As a consequence, it is established that υCpCp(X) has the Lindelöf Σ-property. This answers Problem 47 of Arhangel'skiı̆ (Recent Progress in General Topology, Elsevier Science, 1992). Another consequence is that only the following distribution of the Lindelöf Σ-property is possible in iterated function spaces: (1) Cp,n+1(X) is a Lindelöf Σ-space for every n∈ω; (2) Cp,n+1(X) is a Lindelöf Σ-space only for odd n∈ω; (3) Cp,n+1(X) is a Lindelöf Σ-space only for even n∈ω; (4) for any n∈ω the space Cp,n+1(X) is not a Lindelöf Σ-space.As an application of the developed technique, we prove that, if X is a Tychonoff space such that ω1 is a caliber for X and Cp(X) is a Lindelöf Σ-space, then X has a countable network. This settles Problem 69 of Arhangel'skiı̆ (Recent Progress in General Topology, Elsevier Science, 1992).
Some necessary and some sufficient conditions for Cp(X) and Cp(X,T) being Lindelöf Σ-spaces are obtained. We also get some results on the descriptive complexity of CpX.
We study systematically a class of spaces introduced by G. A. Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, ω-stable and ω-monolithic. It is also established that any Sokolov compact space X is Fréchet-Urysohn and the space C p (X) is Lindelöf. We prove that any Sokolov space with a G δ -diagonal has a countable network and obtain some cardinality restrictions on subsets of small pseudocharacter lying in Σ-products of cosmic spaces.
We consider special subclasses of the class of Lindelöf Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class LΣ(⩽κ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ⩽ω. In the case κ=ω, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tkačenko. The case κ=1 corresponds to the spaces of countable network weight, but even the case κ=2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ1 is in the class LΣ(⩽ω), answering a question of Tkachuk. As well, we study whether certain compact spaces in LΣ(⩽ω) have dense metrizable subspaces, partially answering a question of Tkačenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.
Sixteen years of Cp-theory
- Tkachuk