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Families of continuous retractions and function spaces
Abstract
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.
... We give a new characterization of Valdivia compact spaces: A compact space is Valdivia if and only if it has a dense commutatively monotonically retractable subspace. This result solves Problem 5.12 from [6]. Besides, we introduce the notion of full c-skeleton and prove that a compact space is Corson if and only if it has a full c-skeleton. ...
... Our of main purpose in the first section is to apply a method similar to the one used in [3] to give another proof of this characterization of Valdivia compact spaces. We use this characterization to answer a question in [6], by characterizing Valdivia compact spaces in terms of families of networks and retractions. We know that Corson compact spaces are characterized as compact spaces with a full r-skeleton [5]. ...
... We know that Corson compact spaces are characterized as compact spaces with a full r-skeleton [5]. In [6], function spaces over Corson compact spaces were characterized by using families of R-quotient maps and countable sets (full q-skeletons), in an analogous sense to r-skeletons. In the last part of this paper, we provide a new characterization of Corson compact spaces by using families of closed sets and continuous maps (c-skeletons). ...
We give a new characterization of Valdivia compact spaces: A compact space is Valdivia if and only if it has a dense commutatively monotonically retractable subspace. This result solves Problem 5.12 from \cite{sal-rey}. Besides, we introduce the notion of full c-skeleton and prove that a compact space is Corson if and only if it has a full c-skeleton.
... Let us note that, quite surprisingly, there was independently introduced also the notion of monotonically retractable topological spaces which turned out to be very closely related to the study of compact spaces that admit a retractional skeleton, see [10], and from there on, several results and modifications of the corresponding notions were considered, see e.g. [5,16,17]. ...
We deeply study retractions associated to suitable models in compact spaces admitting a retractional skeleton and find several interesting consequences. Most importantly, we provide a new characterization of Valdivia compacta using the notion of retractional skeletons, which seems to be helpful when characterizing its subclasses. Further, we characterize Eberlein and semi-Eberlein compacta in terms of retractional skeletons and show that our new characterizations give an alternative proof of the fact that continuous image of an Eberlein compact is Eberlein as well as new stability results for the class of semi-Eberlein compacta, solving in particular an open problem posed by Kubis and Leiderman.
We provide simple characterizations of spaces admitting full r-skeletons, c-skeletons and q-skeletons, by using ω-monotone functions. We use this characterizations to prove that every countably compact space admitting a full r-skeleton is proximal; furthermore the characterizations are used to show that the class of spaces admitting full c-skeletons is invariant under subspaces, disjoint topological sums and Σ-products, in addition to prove that the class of spaces admitting full q-skeletons is closed under extensions, continuous images and one-point Lindelöf extensions of disjoint topological sums. These characterizations also yield some positive results for products.
It is know that the Valdivia compact spaces can be characterized by a special family of retractions called r-skeleton (see [10]). Also we know that there are compact spaces with r-skeletons which are not Valdivia. In this paper, we shall study r-skeletons and special families of closed subsets of compact spaces. We prove that if X is a zero-dimensional compact space and {rs:s∈Γ} is an r-skeleton on X such that |rs(X)|≤ω for all s∈Γ, then X has a dense subset consisting of isolated points. Also we give conditions to an r-skeleton in order that this r-skeleton can be extended to an r-skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a Σ-product of a power of the unit interval. Following this fact we introduce the notion of π-skeleton on a compact space X by embedding X in a suitable power of the unit interval together with a pair (F,φ), where F is family of metric separable subspaces of X and φ an ω-monotone function which satisfy certain properties. This new notion generalize the idea of a Σ-product. We prove that a compact space admits a r-skeleton iff it admits a π-skeleton. This equivalence allows to give a new proof of the fact that the product of compact spaces with r-skeletons admits an r-skeleton (see [9]).
It is know that the Valdivia compact spaces can be characterized by a special family of retractions called r-skeleton (see \cite{kubis1}). Also we know that there are compact spaces with r-skeletons which are not Valdivia. In this paper, we shall study r-squeletons and special families of closed subsets of compact spaces. We prove that if X is a zero-dimensional compact space and is an r-skeleton on X such that for all , then X has a dense subset consisting of isolated points. Also we give conditions to an r-skeleton in order that this r-skeleton can be extended to an r-skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a -product of a power of the unit interval. Following this fact we introduce the notion of -skeleton on a compact space X by embedding X in a suitable power of the unit interval together with a pair , where is family of metric separable subspaces of X and an -monotone function which satisfy certain properties. This new notion generalize the idea of a -product. We prove that a compact space admits a retractional-skeleton iff it admits a -skeleton. This equivalence allows to give a new proof of the fact that the product of compact spaces with retractional-skeletons admits an retractional-skeleton (see \cite{cuth1}). In \cite{casa1}, the Corson compact spaces are characterized by a special family of closed subsets. Following this direction, we introduce the notion of weak c-skeleton which under certain conditions characterizes the Valdivia compact spaces and compact spaces with r-skeletons.
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].
An r-skeleton on a compact space is a family of continuous retractions having certain rich properties. The r-skeletons have been used to characterized the Valdivia compact spaces and the Corson compact spaces. Here, we characterized a compact space with an r-skeleton, for which the given r-skeleton can be extended to an r-skeleton on the Alexandroff Duplicate of the given space. Besides, we prove that if X is a zero-dimensional compact space without isolated points and is an r-skeleton on X, then there is such that is not countable.
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.
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.
A consequence of the main proposition includes results of Tacon, and John and Zizler and says: If a Banach space X possesses a continuous Gâteaux differentiable function with bounded nonempty support and with norm-weak continuous derivative, then its dual X* admits a projectional resolution of the identity and a continuous linear one-to-one mapping into c0 (Γ). The proof is easy and selfcontained and does not use any complicated geometrical lemma. If the space X is in addition weakly countably determined, then X* has an equivalent dual locally uniformly rotund norm. It is also shown that l∞ admits no continuous Gâteaux differentiable function with bounded nonempty support.
We prove some generalizations of results concerning Valdivia compact spaces
(equivalently spaces with a commutative retractional skeleton) to the spaces
with a retractional skeleton (not necessarily commutative). Namely, we show
that the dual unit ball of a Banach space is Corson provided the dual unit ball
of every equivalent norm has a retractional skeleton. Another result to be
mentioned is the following. Having a compact space K, we show that K is Corson
if and only if every continuous image of K has a retractional skeleton.
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].
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].
We prove that a countably compact space is monotonically retractable if and
only if it has a full retractional skeleton. In particular, a compact space is
monotonically retractable if and only if it is Corson. This gives an answer to
a question of R. Rojas-Hern{\'a}ndez and V. V. Tkachuk. Further, we apply this
result to characterize retractional skeleton using a topology on the space of
continuous functions, answering thus a question of the first author and a
related question of W. Kubi\'s.
The notion of monotonically monolithic space was introduced by V.V. Tkachuk in 2009 [8]. In this paper we introduce the notion of monotone stability and show that a space Cp(X)Cp(X) is monotonically monolithic if and only if X is monotonically stable. As a consequence, a space Cp(X)Cp(X) is monotonically stable if and only if X is monotonically monolithic. We also prove that Cp(X)Cp(X) is monotonically monolithic when X is a ΣκΣκ-product of a family of Lindelöf Σ-spaces. These results answer some questions posed by Á. Tamariz-Mascarúa and the author in a previous paper [7].
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.
This is a sequel of the work done on (strongly) monotonically monolithic spaces and their generalizations. We introduce the notion of monotonically κ-monolithic space for any infinite cardinal κ and present the relevant results. We show, among other things, that any σ-product of monotonically κ-monolithic spaces is monotonically κ-monolithic for any infinite cardinal κ; besides, it is consistent that any strongly monotonically ω-monolithic space with caliber ω
1 is second countable. We also study (strong) monotone κ-monolithicity in linearly ordered spaces and subspaces of ordinals.
We characterize Corson-compact spaces by means of countable elementary sub- structures.
We apply elementary substructures to characterize the spaceCp(X) for Corson- compact spaces. As a result, we prove that a compact space X is Corson-compact, if Cp(X) can be represented as a continuous image of a closed subspace of (L�) ! × Z, where Z is compact and Ldenotes the canonical Lindelof space of cardinalitywith one non-isolated point. This answers a question of Archangelskij (2).
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 collect examples of Valdivia compact spaces, their continuous images and associated classes of Banach spaces which appear naturally in various branches of mathematics. We focus on topological constructions generating Valdivia compact spaces, linearly ordered compact spaces, compact groups, L1 spaces, Banach lattices and noncommutative L1 spaces.
We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of retractions of a certain kind. Consequently, a compact space of weight⩽ℵ1 is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight⩽ℵ1 is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.
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E-mail address: sgarcia@matmor.unam.mx E-mail address: satzchen@yahoo
- Centro De
- Ciencias Matemáticas
Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Campus
Morelia, Apartado Postal 61-3, Santa María, 58089, Morelia, Michoacán, México.
E-mail address: sgarcia@matmor.unam.mx
E-mail address: satzchen@yahoo.com.mx