Hereditary invertible linear surjections and splitting problems for selections

Let A+B be the pointwise (Minkowski) sum of two convex subsets A and B of a Banach space. Is it true that every continuous mapping h:X→A+B splits into a sum h=f+g of continuous mappings f:X→A and g:X→B? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear surjections between Banach spaces. Some affirmative and negative results on such invertibility with respect to an appropriate class of convex compacta are presented. As a corollary, a positive answer to the above question is obtained for strictly convex finite-dimensional precompact spaces.     

Properties that are productive,closed-hereditary and surjective

We obtain a characterization of all those topological properties of regular Hausdorff spaces, that are preserved under the formation of arbitrary products, closed subspaces and continuous surjections.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

Right inverses of linear maps on convex sets

We study, via continuous selections of multivalued maps, the problem of finding a right inverse to the restriction of a linear map to a convex body.     

Extension of continuous functions in digital spaces with the Khalimsky topology

The digital space Zn equipped with Efim Khalimsky's topology is a connected space. We study continuous functions Zn⊃A→Z, from a subset of Khalimsky n-space to the Khalimsky line. We give necessary and sufficient condition for such a function to be extendable to a continuous function Zn→Z. We classify the subsets A of the digital plane such that every continuous function A→Z can be extended to a continuous function on the whole plane.     

Local connectedness and extension of uniformly continuous functions

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. The straight spaces have been studied in [A. Berarducci, D. Dikranjan, J. Pelant, An additivity theorem for uniformly continuous functions, Topology and its Applications 146-147 (2005) 339-352], which contains characterization of the straight spaces within the class of the locally connected spaces (they are the uniformly locally connected ones) and the class of the totally disconnected spaces (they coincide with the totally disconnected Atsuji spaces). We show that the completion of a straight space is straight and we characterize the dense straight subspaces of a straight space. In order to clarify further the relation between straightness and the level of local connectedness of the space we introduce two more intermediate properties between straightness and uniform local connectedness and we give various examples to distinguish them. One of these properties coincides with straightness for complete spaces and provides in this way a useful characterization of complete straight spaces in terms of the behaviour of the quasi-components of the space.     

Open mappings looking like projections

Every open continuous mappingf from a metric space (X, d) onto a countable-dimensional metric spaceY admits a special type of factorization (Y×[0, 1] throughout), provided all fibers off are dense in itself and complete with respect tod. On this basis, an upper semi-continuous Cantor bouquet of disjoint usco selections for a class of 1.s.c. mappings between metrizable spaces is constructed.     

Compact spaces generated by retractions

We study compact spaces which are obtained from metric compacta by iterating the operation of inverse limit of continuous sequences of retractions. This class, denoted by R, has been introduced in [M. Burke, W. Kubi?, S. Todor?evi?, Kadec norms on spaces of continuous functions, http://arxiv.org/abs/math.FA/0312013]. Allowing continuous images in the definition of class R, one obtains a strictly larger class, which we denote by RC. We show that every space in class RC is either Corson compact or else contains a copy of the ordinal segment ω₁+1. This improves a result of Kalenda from [O. Kalenda, Embedding of the ordinal segment [0,ω₁] into continuous images of Valdivia compacta, Comment. Math. Univ. Carolin. 40 (4) (1999) 777-783], where the same was proved for the class of continuous images of Valdivia compacta. We prove that spaces in class R do not contain cutting P-points (see the definition below), which provides a tool for finding spaces in RC?R. Finally, we study linearly ordered spaces in class RC. We prove that scattered linearly ordered compacta belong to RC and we characterize those ones which belong to R. We show that there are only 5 types (up to order isomorphism) of connected linearly ordered spaces in class R and all of them are Valdivia compact. Finally, we find a universal pre-image for the class of all linearly ordered Valdivia compacta.     

The Namioka property of KC-functions and Kempisty spaces

A topological space Y is called a Kempisty space if for any Baire space X every function , which is quasi-continuous in the first variable and continuous in the second variable has the Namioka property. Properties of compact Kempisty spaces are studied in this paper. In particular, it is shown that any Valdivia compact is a Kempisty space and the Cartesian product of an arbitrary family of compact Kempisty spaces is a Kempisty space.     

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 , 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 Cp(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 Cp(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 Vω into Cp(X) or Cp(X,2) is characterized in terms of projective properties of X.     

A note on a set-valued extension property

Principal result: Suppose Y is metrizable. Then: (a) if X is metrizable and A⊂X is closed, then every continuous g:A→Y extends to an l.s.c. ψ:X→K(Y); (b) Y satisfies (a) for all paracompact X if and only if Y is completely metrizable.     

The compact-open topology: A new perspective

This paper studies the compact-open topology on the set KC(X) of all real-valued functions defined on a Tychonoff space, which are continuous on compact subsets of X. In addition to metrizability, separability and second countability of this topology on KC(X), various kinds of topological properties of this topology are studied in detail. Actually the motivation for studying the compact-open topology on KC(X) lies in the attempt of having a simpler proof for the characterization of a completeness property of the compact-open topology on C(X), the set of all real-valued continuous functions on X.     

Lifting paths on quotient spaces

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn-Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g∈G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.     

Trees and continuous mappings into the real line

Trees of height ω₁ are characterized in terms of continuous mappings to the real line. In particular Souslin trees are characterized as those uncountable trees with no uncountable continuous image in the real line. Trees which can not be continuously embeded in the real line are also characterized.     

Even continuity and topological equicontinuity in topologized semigroups

A topologized semigroup X having an evenly continuous resp., topologically equicontinuous, family RX of right translations is investigated. It is shown that: (1) every left semitopological semigroup X with an evenly continuous family RX is a topological semigroup, (2) a semitopological group X is a paratopological group if and only if the family RX is evenly continuous and (3) a semitopological group X is a topological group if and only if the family RX is topologically equicontinuous. In particular, we get that for any paratopological group X which is not a topological group, the family RX provides an example of a transitive group of homeomorphisms of X that is evenly continuous and not topologically equicontinuous. The last conclusion answers negatively a question posed by H.L. Royden.     

Continuous weak selections for products

A weak selection on an infinite set X   is a function σ:[X]²→X$\sigma :{[X]}^2\to X$ such that σ({x,y})∈{x,y}$\sigma (\{x,y\})\in \{x,y\}$ for each {x,y}∈[X]²$\{x,y\}\in {[X]}^2$. A weak selection on a space is said to be continuous if it is a continuous function with respect to the Vietoris topology on [X]²${[X]}^2$ and the topology on X  . We study some topological consequences from the existence of a continuous weak selection on the product X×Y$X\times Y$ for the following particular cases:

(i)

Both X and Y are spaces with one non-isolated point.     

On the density of the space of continuous and uniformly continuous functions

For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent.     

Metrizable fiberings of continuous images of the lexicographic square

A Tychonoff space A is metrizably fibered if and only if there exists a continuous map onto a metrizable space B such that for each b∈B,F⁻¹(b) is metrizable. We resolve a question stated by V. Tkachuk by showing that every first countable Hausdorff continuous image of the lexicographic square is metrizably fibered. We also observe that an example of S. Mardeši? and P. Papi? resolves a related question stated by Tkachuk.     

Small Valdivia compact 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.     

Products of straight spaces

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X×Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds:

(a)

both X and Y are precompact;

(b)

both X and Y are locally connected;

(c)

one of the spaces is both precompact and locally connected.

In particular, when X satisfies (c), the product X×Z is straight for every straight space Z.Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.