Existence and relaxation theorems for extreme continuous selectors of multifunctions with decomposable values

We present some extreme continuous selector theorems, synthesizing the author's results; namely, we study existence and properties of continuous selectors from the set of extreme points of multifunctions with closed convex decomposable values in the space of Bochner integrable functions.     

Approximations of relations by continuous functions

Let X be a Tychonoff space, C(X) be the space of all continuous real-valued functions defined on X and CL(X×R) be the hyperspace of all nonempty closed subsets of X×R. We prove the following result. Let X be a countably paracompact normal space. The following are equivalent: (a) dimX=0; (b) the closure of C(X) in CL(X×R) with the Vietoris topology consists of all F∈CL(X×R) such that F(x)≠∅ for every x∈X and F maps isolated points into singletons; (c) each usco map which maps isolated points into singletons can be approximated by continuous functions in CL(X×R) with the locally finite topology. From the mentioned result we can also obtain the answer to Problem 5.5 in [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] and to Question 5.5 in [R.A. McCoy, Comparison of hyperspace and function space topologies, Quad. Mat. 3 (1998) 243-258] in the realm of normal, countably paracompact, strongly zero-dimensional spaces. Generalizations of some results from [L'. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173-2182] are also given.     

On continuous extensions

We consider various possiblities concerning the continuous extension of continuous functions taking values in an ultrametric space. In Section 1 we consider Tietze-type exetension theorems concerning continuous extendibility of continuous functions from compact and closed subsets to the whole space. In Sections 2 and 3 we consider extending separated continous functions in such a way that, certain continuous extensions remain separated. Functgions taking values in a complete ultravalued field are dealt with in Section 2, and the real and complex cases in Section 3.     

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.     

Axial continuous functions

A function from the plane to the plane is axial if it does not change one coordinate. We show that not every continuous function can be approximated by a superposition of continuous axial functions. This is a counterexample to a possible generalization of theorem of Eggleston about continuous bijections.     

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.     

Sequences of semicontinuous functions accompanying continuous functions

A space X is said to have property (USC) (resp. (LSC)) if whenever is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval [0,1] converging pointwise to the constant function 0 with the value 0, there is a sequence of continuous functions from X into [0,1] such that fn?gn (n∈ω) and converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepers' conjecture on properties S₁(Γ,Γ) and wQN.     

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.     

Characterizing continuous functions on compact spaces

We consider the following problem: given a set X and a function , does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we also characterize homeomorphisms on compact metric spaces.     

Modifications of sequence selection principles

We study modifications of the sequence selection principles for real functions obtained by allowing other than continuous functions and/or by replacing the pointwise convergence by quasi-normal or discrete convergence. We show that for important families of real functions we obtain already known sequence selection principles. Replacing the pointwise convergence we obtain plenty of sequence selection principles and we try to find equivalences or implications among them. Finally we attempt to show that these modifications are useful tools for proofs of known results or their strengthening.     

Spaces of continuous functions over a Ψ-space

The Lindelöf property of the space of continuous real-valued continuous functions is studied. A consistent example of an uncountable Ψ-like space is constructed for which the space of continuous real-valued functions with the pointwise convergence topology is Lindelöf.     

General insertion and extension theorems for localic real functions

In this paper we investigate localic real functions on frames. We provide a necessary and sufficient condition for the insertion of a continuous localic real function between two arbitrary comparable localic real functions. We also establish necessary and sufficient conditions for extending a bounded localic real function from a complemented sublocale to the whole frame.     

A study concerning splitting and jointly continuous topologies on C(Y,Z)

Let Y and Z be two fixed topological spaces and C(Y, Z) the set of all continuous maps from Y into Z. We construct and study topologies on C(Y, Z) that w^{e call F}n(τn)-family-open topologies. Furthermore, we find necessary and su?cient conditions such that these topologies to be splitting and jointly continuous. Finally, we present questions concerning a further study on this area.     

On the continuity of separately continuous bihomomorphisms

Separately continuous bihomomorphisms on a product of convergence or topological groups occur with great frequency. Of course, in general, these need not be jointly continuous. In this paper, we exhibit some results of Banach-Steinhaus type and use these to derive joint continuity from separate continuity. The setting of convergence groups offers two advantages. First, the continuous convergence structure is a powerful tool in many duality arguments. Second, local compactness and first countability, the usual requirements for joint continuity, are available in much greater abundance for convergence groups.     

Most continuous descent methods converge

We consider continuous descent methods for the minimization of convex functions defined on a general Banach space and show that most of them (in the sense of Baire category) converge.Received: 21 July 2004     

Clean rings of continuous functions

A ring is clean if every element is the sum of a unit and an idempotent. Let be a dense local subring of the reals which is not a field. We show that the ring of A-valued continuous functions on a zero-dimensional space X is clean if and only if X is a P-space, and examine some properties of the prime ideal spectrum of this ring. Received June 3, 2005; accepted in final form December 3, 2005.     

A Banach space where minimal weak cuscos are generically compact-valued

The space of continuous functions on the double arrow space has long been of interest in differentiability theory since many convex functions on this space are densely but not generically Gâteaux differentiable. We show that this space has the property that minimal weak* cuscos into its dual take compact values at the points of a denseG set.     

Convergence theorems for continuous descent methods

We examine continuous descent methods for the minimization of Lipschitzian functions defined on a general Banach space. We establish several convergence theorems for those methods which are generated by regular vector fields. Since the complement of the set of regular vector fields is -porous, we conclude that our results apply to most vector fields in the sense of Baires categories.     

Implications of pseudo-orbit tracing property for continuous maps on compacta

We look at the dynamics of continuous self-maps of compact metric spaces possessing the pseudo-orbit tracing property (i.e., the shadowing property). Among other things we prove the following: (i) the set of minimal points is dense in the non-wandering set Ω(f), (ii) if f has either a non-minimal recurrent point or a sensitive minimal subsystem, then f has positive topological entropy, (iii) if X is infinite and f is transitive, then f is either an odometer or a syndetically sensitive non-minimal map with positive topological entropy, (iv) if f has zero topological entropy, then Ω(f) is totally disconnected and f restricted to Ω(f) is an equicontinuous homeomorphism.     

Topological continuous convergence

Let X be a limit space, Y a topological space. We show that c(X,Y), the limitierung of continuous convergence on LIM(X,Y), is topological whenever X is basic locally compact. For regular Y, local compactness of X is sufficient. In both cases, c(X,Y) coincides with the compact-open topology. If X satisfies a certain regularity condition, the fact that c(X,Y) is topological implies, conversely, that X is (basic) locally compact.The author would like to thank S. Weck for some inspiring discussions.