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.     

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.     

A separation theorem for semicontinuous functions on preordered topological spaces

Suppose that X is a topological space with preorder , and that –g, f are bounded upper semicontinuous functions on X such that g(x) f(y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f, and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied.Received: 26 May 2004     

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.     

Function spaces and one point extensions for the construct of metered spaces

The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M.     

Isometric embeddings of real projective spaces into Euclidean spaces

This paper studies isometric embeddings of RPn via non-degenerate symmetric bilinear maps. The main result shows the infimum dimension of target Euclidean spaces among these constructions for RPn is . Next, we construct Veronese maps by induction, which realize the infimum. Finally, we give a simple proof of Rigidity Theorem of Veronese maps.     

Group action on zero-dimensional spaces

Let X be a Tychonoff space, H(X) the group of all self-homeomorphisms of X and the evaluation function. Call an admissible group topology on H(X) any topological group topology on H(X) that makes the evaluation function a group action. Denote by LH(X) the upper-semilattice of all admissible group topologies on H(X) ordered by the usual inclusion. We show that if X is a product of zero-dimensional spaces each satisfying the property: any two non-empty clopen subspaces are homeomorphic, then LH(X) is a complete lattice. Its minimum coincides with the clopen-open topology and with the topology of uniform convergence determined by a T₂-compactification of X to which every self-homeomorphism of X continuously extends. Besides, since the left, the right and the two-sided uniformities are non-Archimedean, the minimum is also zero-dimensional. As a corollary, if X is a zero-dimensional metrizable space of diversity one, such as for instance the rationals, the irrationals, the Baire spaces, then LH(X) admits as minimum the closed-open topology induced by the Stone-?ech-compactification of X which, in the case, agrees with the Freudenthal compactification of X.     

Note on function spaces with the topology of pointwise convergence

The note contains two examples of function spaces C _p (X) endowed with the pointwise topology. The first example is C _p (M), M being a planar continuum, such that C _p (M) ^m is uniformly homeomorphic to C _p (M) ⁿ if and only if m = n. This strengthens earlier results concerning linear homeomorphisms. The second example is a non-Lindelöf function space C _p (X), where X is a monolithic perfectly normal compact space all linearly orderable closed subspaces of which are metrizable. This example is obtained under the additional set-theoretical axiom . This solves a problem of Arhangelskiĭ.     

Injective spaces via adjunction

The work of the present author and his coauthors over the past years gives evidence that it may be useful to regard each topological space as a kind of enriched category, by interpreting the convergence relation x→x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from enriched Category Theory for the investigation of topological spaces. Topological theories introduced by the author provide a convenient general setting for appropriately transferring these concepts and ideas to the world of topological spaces and some other geometric objects such as approach spaces. Using tools like adjunction and the Yoneda lemma, we show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on . This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.     

Function spaces in metrically generated theories

The theory of metrically generated constructs provides us with an excellent setting for the study of function spaces. In this paper we develop a function space theory for metrically generated constructs and, by considering different metrically generated constructs, we capture interesting examples. For instance, for uniform spaces we retrieve the uniformity of uniform convergence and its generalization to Σ-convergence and for UG-spaces we obtain a quantified version of these structures. Our theory also allows for many applications, in particular we are able to characterize the complete subspaces of these function spaces and we succeed in producing an appropriate Ascoli theorem.     

Dually discrete spaces

A neighbourhood assignment in a space X is a family of open subsets of X such that x∈Ox for any x∈X. A set Y⊆X is a kernel ofO if . We obtain some new results concerning dually discrete spaces, being those spaces for which every neighbourhood assignment has a discrete kernel. This is a strictly larger class than the class of D-spaces of [E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (2) (1979) 371-377].     

On metric spaces and local extrema

The main aim of this paper is to give a positive answer to a question of Behrends, Geschke and Natkaniec regarding the existence of a connected metric space and a non-constant real-valued continuous function on it for which every point is a local extremum. Moreover we show that real-valued continuous functions on connected spaces such that every family of pairwise disjoint non-empty open sets is of size <|R| are constant provided that every point is a local extremum.     

Local connectivity functions on arcwise connected spaces and certain continua

Types of spaces are given on which every local connectivity function is a connectivity function, a connected function, or a Darboux function. A complete determination such spaces is obtained when the spaces are assumed to be arc-like continua or circle-like continua. Results provide answers to a question asked by Stallings.     

Mappings and Prohorov spaces

We consider completely regular Hausdorff spaces. In this paper we investigate the space of probability Radon measures P(X) on a space X and the property to be a Prohorov space. We prove that the space P(X) is sieve-complete if and only if X is sieve-complete. Every mapping generates the mapping . Some properties of the mapping P(φ) are studied. In particular, we investigate under which conditions the open continuous image of a Prohorov space is Prohorov.     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

Remarks on star-C-Menger spaces

In this paper, we show that assuming ω₁ < , there exists a Tychono? star-C-Menger space having a regular-closed G_δ-subspace which is not star-C-Menger which gives a partial answer to a question of Song and Yin [12], and continue to investigate topological properties of star-C-Menger spaces.     

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.     

Multistage n-dimensional universal spaces and extensions

Let Iτ be the Tychonoff cube of weight τ?ω with a fixed point, στ and Στ be the correspondent σ- and Σ-products in Iτ and στ⊂(Σστ=ω(στ))⊂Στ. Then for any n∈{0,1,2,…}, there exists a compactum Unτ⊂Iτ of dimension n such that for any Z⊂Iτ of dimension?n, there exists a topological embedding of Z in Unτ that maps the intersections of Z with στ, Σστ and Στ to the intersections , and of Unτ with στ, Σστ and Στ, respectively; , and are n-dimensional and is σ-compact, is a Lindelöf Σ-space and is a sequentially compact normal Fréchet-Urysohn space. This theorem (on multistage universal spaces of given dimension and weight) implies multistage extension theorems (in particular, theorems on Corson and Eberlein compactifications) for Tychonoff spaces.     

The evaluation identification in function spaces

A study is made of the natural function which maps each point x of a space X to the evaluation function e_x:Y^x→Y defined by $\text{e}{}_{\mathrm{x}}\text{(?)=?(x)}$. A consequence of the results is that βX and υX can both be considered as subspaces of spaces of continous functions from appropriate domain spaces into I or R, respectively.     

Distances from selectors to spaces of Baire one functions

Given a metric space X and a Banach space (E,‖⋅‖) we study distances from the set of selectors Sel(F) of a set-valued map to the space B₁(X,E) of Baire one functions from X into E. For this we introduce the d-τ-semioscillation of a set-valued map with values in a topological space (Y,τ) also endowed with a metric d. Being more precise we obtain that