Continuous selections avoiding extreme points

It is shown that if X is a countably paracompact collectionwise normal space, Y is a Banach space and φ:X→Y² is a lower semicontinuous mapping such that φ(x) is Y or a compact convex subset with Cardφ(x)>1 for each x∈X, then φ admits a continuous selection f:X→Y such that f(x) is not an extreme point of φ(x) for each x∈X. This is an affirmative answer to the problem posed by V. Gutev, H. Ohta and K. Yamazaki [V. Gutev, H. Ohta and K. Yamazaki, Selections and sandwich-like properties via semi-continuous Banach-valued functions, J. Math. Soc. Japan 55 (2003) 499-521].     

Ordinals and set-valued zero-selections for hyperspaces

A continuous zero-selection f for the Vietoris hyperspace F(X) of the nonempty closed subsets of a space X is a Vietoris continuous map f:F(X)→X which assigns to every nonempty closed subset an isolated point of it. It is well known that a compact space X has a continuous zero-selection if and only if it is an ordinal space, or, equivalently, if X can be mapped onto an ordinal space by a continuous one-to-one surjection. In this paper, we prove that a compact space X has an upper semi-continuous set-valued zero-selection for its Vietoris hyperspace F(X) if and only if X can be mapped onto an ordinal space by a continuous finite-to-one surjection.     

Whitney preserving maps on finite graphs

For a Whitney preserving map f:X→G we show the following: (a) If X is arcwise connected and G is a graph which is not a simple closed curve, then f is a homeomorphism; (b) If X is locally connected and G is a simple closed curve, then X is homeomorphic to either the unit interval [0,1], or the unit circle S¹. As a consequence of these results, we characterize all Whitney preserving maps between finite graphs. We also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.     

On the existence of continuous (approximate) roots of algebraic equations

The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm ∥⋅_∥∞. The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.Also we consider continuous approximate roots of the equation zn−f=0 with respect to z, where f∈C(X), and provide a topological characterization of compact Hausdorff space X with dimX?1 such that the above equation has an approximate root in C(X) for each f∈C(X), in terms of the first ?ech cohomology of X.     

Function spaces from core compact coherent spaces to continuous B-domains

It is proved in this paper that for a continuous B-domain L, the function space [X→L] is continuous for each core compact and coherent space X. Further, applications are given. It is proved that:

(1)

the function space from the unit interval to any bifinite domain which is not an L-domain is not Lawson compact;

(2)

the Isbell and Scott topologies on [X→L] agree for each continuous B-domain L and core compact coherent space X.

     

Expansive canonical coordinates are hyperbolic

Theorem. Let ?:X→X be an expansive homeomorphism of a compact metric space onto itself and let ? have canonical coordinates. Then there exists a metric compatible with the topology of X with respect to which the canonical coordinates are hyperbolic.     

Generic uniqueness of equilibrium points

Let X be a nonempty, convex and compact subset of normed linear space E (respectively, let X be a nonempty, bounded, closed and convex subset of Banach space E and A be a nonempty, convex and compact subset of X) and f:X×X→R be a given function, the uniqueness of equilibrium point for equilibrium problem which is to find x^∗∈X (respectively, x^∗∈A) such that f(x^∗,y)≥0 for all y∈X (respectively, f(x^∗,y)≥0 for all y∈A) is studied with varying f (respectively, with both varying f and varying A). The results show that most of equilibrium problems (in the sense of Baire category) have unique equilibrium point.     

Minimal sets in compact connected subspaces

Let X be a topological space, f:X→X be a continuous map, and Y be a compact, connected and closed subset of X. In this paper we show that, if the boundary X_∂Y contains exactly one point v and f(v)∈Y, then Y contains a minimal set of f.     

On the root closedness of continuous function algebras

For a compact Hausdorff space X, C(X) denotes the algebra of all complex-valued continuous functions on X. For a positive integer n, we say that C(X) is n-th root closed if, for each f∈C(X), there exists g∈C(X) such that f=gn. It is shown that, for each integer m?2, there exists a compact Hausdorff space Xm such that C(Xm) is m-th root closed, but not n-th root closed for each integer n relatively prime to m. This answers a question posed by Countryman Jr. [R.S. Countryman Jr., On the characterization of compact Hausdorff X for which C(X) is algebraically closed, Pacific J. Math. 20 (1967) 433-438] et al.     

Domain representability and the Choquet game in Moore and BCO-spaces

In this paper we investigate the role of domain representability and Scott-domain representability in the class of Moore spaces and the larger class of spaces with a base of countable order. We show, for example, that in a Moore space, the following are equivalent: domain representability; subcompactness; the existence of a winning strategy for player α (= the nonempty player) in the strong Choquet game Ch(X); the existence of a stationary winning strategy for player α in Ch(X); and Rudin completeness. We note that a metacompact ?ech-complete Moore space described by Tall is not Scott-domain representable and also give an example of ?ech-complete separable Moore space that is not co-compact and hence not Scott-domain representable. We conclude with a list of open questions.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

Some results on separate and joint continuity

Let f:X×K→R be a separately continuous function and C a countable collection of subsets of K. Following a result of Calbrix and Troallic, there is a residual set of points x∈X such that f is jointly continuous at each point of {x}×Q, where Q is the set of y∈K for which the collection C includes a basis of neighborhoods in K. The particular case when the factor K is second countable was recently extended by Moors and Kenderov to any ?ech-complete Lindelöf space K and Lindelöf α-favorable X, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when K is a Lindelöf p-space and X is conditionally σ-α-favorable space. Here we add new results of this sort when the factor X is σC(X)-β-defavorable and when the assumption “base of neighborhoods” in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.     

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.     

On Peano's theorem in Banach spaces

We show that if X is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping f:X→X such that the autonomous differential equation x^′=f(x) has no solution at any point.     

A non-separable Christensen's theorem and set tri-quotient maps

For every space X let K(X) be the set of all compact subsets of X. Christensen [J.P.R. Christensen, Necessary and sufficient conditions for measurability of certain sets of closed subsets, Math. Ann. 200 (1973) 189-193] proved that if X,Y are separable metrizable spaces and F:K(X)→K(Y) is a monotone map such that any L∈K(Y) is covered by F(K) for some K∈K(X), then Y is complete provided X is complete. It is well known [J. Baars, J. de Groot, J. Pelant, Function spaces of completely metrizable space, Trans. Amer. Math. Soc. 340 (1993) 871-879] that this result is not true for non-separable spaces. In this paper we discuss some additional properties of F which guarantee the validity of Christensen's result for more general spaces.     

On metric spaces with the properties of de Groot and Nagata in dimension one

A metric space (X,d) has the de Groot property GPn if for any points x₀,x₁,…,xn+2∈X there are positive indices i,j,k?n+2 such that i≠j and d(xi,xj)?d(x₀,xk). If, in addition, k∈{i,j} then X is said to have the Nagata property NPn. It is known that a compact metrizable space X has dimension dim(X)?n iff X has an admissible GPn-metric iff X has an admissible NPn-metric.We prove that an embedding f:(0,1)→X of the interval (0,1)⊂R into a locally connected metric space X with property GP₁ (resp. NP₁) is open, provided f is an isometric embedding (resp. f has distortion Dist(f)=‖fLip_‖⋅‖f−1Lip_‖<2). This implies that the Euclidean metric cannot be extended from the interval [−1,1] to an admissible GP₁-metric on the triode T=[−1,1]∪[0,i]. Another corollary says that a topologically homogeneous GP₁-space cannot contain an isometric copy of the interval (0,1) and a topological copy of the triode T simultaneously. Also we prove that a GP₁-metric space X containing an isometric copy of each compact NP₁-metric space has density ?c.     

Cut points in some connected topological spaces

We prove that a connected topological space with endpoints has exactly two non-cut points and every cut point is a strong cut point; it follows that such a space is a COTS and the only two non-cut points turn out to be endpoints (in each of the two orders) of the COTS. A non-indiscrete connected topological space with exactly two non-cut points and having only finitely many closed points is proved homeomorphic to a finite subspace of the Khalimsky line. Further, it is shown, without assuming any separation axiom, that in a connected and locally connected topological space X, for a, b in X, S[a,b] is compact whenever it is closed. Using this result we show that an H(i) connected and locally connected topological space with exactly two non-cut points is a compact COTS with end points.     

A generalisation of the functional approach to compactness

A topological space X is compact iff the projection π:X×Y→Y is closed for any space Y. Taking this as a definition and then asking that π maps α-closed subspaces of X×Y onto β-closed subspaces of Y, for different closures α and β, extends the notion of compactness to include also examples of “asymmetric compactness” pursued in the article.Categorical closure operators and a so-called “functional approach to general topology” are employed to define and prove fundamental properties of compact objects and proper maps in this generalised setting.     

A class of functions whose sublevel sets are absolute retracts

In this paper, we prove a result of which the following is a corollary: If X is a Banach space and J:X→R is a contraction, then the nonempty sublevel sets of the function x→‖x‖+J(x) are absolute retracts.