Fell topology on the space of functions with closed graph

LetX andY beT ₁ topological spaces andG(X, Y) the space of all functions with closed graph. Conditions under which the Fell topology and the weak Fell topology coincide onG(X,Y) are given. Relations between the convergence in the Fell topologyτF, Kuratowski and continuous convergence are studied too. Characterizations of a topological spaceX by separation axioms of (G(X, R), τF) and topological properties of (G(X, R), τF) are investigated.     

Spaces of Densely Continuous Forms

The set C(X,Y) of continuous functions from a topological space X into a topological space Y is extended to the set D(X,Y) of densely continuous forms from X to Y, such form being a kind of multifunction from X to Y. The topologies of pointwise convergence, uniform convergence, and uniform convergence on compact sets are defined for D(X,Y), for locally compact spaces X and metric spaces Y having a metric satisfying the Heine–Borel property. Under these assumptions, D(X,Y) with the uniform topology is shown to be completely metrizable. In addition, if X is compact, D(X,Y) is completely metrizable under the topology of uniform convergence on compact sets. For this latter topology, an Ascoli theorem is established giving necessary and sufficient conditions for a subset of D(X,Y) to be compact.     

The bounded-open topology on function spaces

For any regular space Z It is shown, 1) that the bounded-open topology T on C(Y,Z) is splitting and it is also the smallest jointly continuous topology whenever Y is locally bounded, 2) if Y is locally bounded or if X × Y is a boundedly generated space, then there is a natural bijection on C(X × Y,Z) onto C(X,(C(Y,Z),T_eo) which is actually a homeomorphism with respect to the bounded-open topology on both function spaces, 3) The path components of (C(Y,Z),T_eo) are exactly its homotopy classes whenever Y is boundedly generated, 4) The bounded-open topology Teo induces contravariant and covariant Homotopy preserving function-space functors. Further, 5) T_eo reduces to the compact-open topology t_co whenever the domain Y is regular; but in general, T_eo is finer than T_co (assuming the domain is Hausdorff or the range is either Hausdorff or regular).     

THE INDUCED H-STRUCTURE ON FUNCTION SPACES

Abstract

Let (Z,Γ) be an H-structure. Then, for each exponential object Y in TOP, an H-structure is induced on the topological space C_t(Y,Z) of continuous maps equipped with the appropriate function space topology t (e.g. t = T_is, where T_is is the Isbell topology on C(Y,Z)).

If (Z,Γ) is H-associative (resp.admits inversion), then the induced H-structure is also H-associative (resp. admits inversion).

If (Z,Γ) is H-associative and admits inversion (e.g. a topological group) then all path components of C_t(Y,Z) belong to the same homotopy type.

We also study the special case of (Z,Γ) being a topological group. Moreover, we prove that certain functions between function spaces are H-homomorphisms of the induced H-structures in the function spaces.     

Compactness for a class of hit-and-miss hyperspaces

In the study of some kind of generalized Vietoris-type topologies for the hyperspace of all nonempty closed subsets of a topological space (X, τ), namely the so called Δ-hit-and-miss-topologies with Δ⊇Cl(X) (or Δ-topologies), which was initiated by the second author in 1965, it is obvious, that the non-compactness of such a hyperspace often depends on the non-compactness even in the lower-semifinite topology (induced by the “hit-sets”), which is contained in all hypertopologies of this type. Otherwise, compactness for these topologies is easily obtained from the compactness of (X, τ) by well-known theorems, if the “miss-sets” are induced either by compact or closed subsets. To obtain a similar result for topologies with “miss-sets” generated by subsets with a property which generalizes both, closedness and compactness especially in the non-Hausdorff case, we use consequently a quite set-theoretical lemma, stated at the beginning.     

Characterization of the Function Classes C(Y)|X

Let X be a topological space and let Φ ? C(X). Then there exists a topological space Y containing X as a subspace and such that Φ = C(Y)¦X, if and only if Φ is weakly composition closed, i.e., for any index set I, any fi ∈ Φ (i ∈ I) and any continuous map k : RI → Rwe have k ° 〈fi〉 ∈ Φ, where 〈fi〉 : X → RI is the map with i-th coordinate fi. The analogous statement is valid for functions to any T1 space, rather than to R, and even we can consider functions to any set of T1 spaces, and then a generalization of the above statement is valid, with a suitably defined weak composition closedness property. We also show that some earlier results on characterization of function classes Φ ? C(X) of the form C(Y)¦X, with Y some extension of a given topological space X, and on the characterization of function classes C(〈X, T〉), with T some topology on a given set X, respectively, can be generalized in an analogous way as above, by means of composition properties analogous to the above one or by filter closedness (for functions to any set of T3 spaces, or to any set of topological spaces, respectively).     

Some topologies on a Šerstnev probabilistic normed space

In this paper we will introduce two other topologies, coarser than the so-called strong topology, on a class of Šerstnev probabilistic normed spaces, and obtain some important properties of these topologies. We will show that under the first topology, denoted by τ₀, our probabilistic normed space is decomposable into the topological direct sum of a normable subspace and the subspace of probably null elements. Under the second topology, which is in fact the inductive limit topology of a family of locally convex topologies, the dual space becomes a locally convex topological vector space.     

集值映射空间在紧开拓扑下的N性质(英文)

本文研究了点紧连续集值映射族在紧开拓扑下的N性质.利用cs-σ网方法获得了如下结果:若X是N0空间,Y是N空间,则C_k(X,Y)是N空间.该结论将J.A.Guthrie关于单值连续映射空间的结论推广到了集值映射空间上,并且改进了相关结论.     

Range universal spaces

We characterize those topological spaces Y for which the Isbell and finest splitting topologies on the set C(X,Y) of all continuous functions from X into Y coincide for all topological spaces X. We also consider the same question for the coincidence of the restriction of the finest splitting topology on the upper semicontinuous set-valued functions to C(X,Y) and the finest splitting topology on C(X,Y). In the first case, the spaces in question are, after identifying points that are in each others closures, subsets of the two point Sierpiński space, which gives a converse and generalization of a result of S. Dolecki, G.H. Greco, and A. Lechicki. In the second case, the spaces in question are, after identifying points that are in each others closures, order bases for bounded complete continuous DCPOs with the Scott topology.     

Bitopologies on products and ratios

The purpose of the paper is to extend the notions of splitting and jointly continuous topologies on function spaces to products of spaces, to present a new theory of duality between topologies on function spaces and topologies on products of spaces, and to generalize these theories to bitopological spaces. The author starts from the definitions well known in the literature. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 242, 1997, pp. 217–229. Translated by N. Yu. Netsvetaev.     

类W-KKM(X,Y,Z),几乎不动点定理和不动点定理

引进了相对弱$R$-子集和类($W$-)KKM$(X,Y,Z)$的概念,给出了相对KKM映射与相对弱$R$-子集之间的等价关系以及$W$-KKM$(X,Y,Z)$的一个性质,然后给出了两个连续选择定理并得到不动点定理和重合点定理, 最后,在一致拓扑空间上得到具有弱-KKM性质的映射的几乎不动点,不动点和重合点的存在定理.     

Nearly exotic topologies on normed spaces

It is proved that for every infinite dimensional normed space (E, ‖ ‖) there is a non-trivial linear space topologyτ onE which is weaker than the norm topology and is such that (E, τ) admits no non-trivial continuous linear functionals. IfE is a space with a generalized basis or is aC(X) space, it is proved that the topologyτ can be taken to be Hausdorff.     

Computation of the homology ofΩ(X∨Y)

Summary We compute the homology of Ω(X∨Y) (the loop space of the wedge of the spaces X and Y), in terms of the homogies of ΩX and ΩY. To do this we use the fact that our problem is equivalent to the computation of the homology of the free product of two topological groups in terms of the homologies of the topological groups. We establish a multiple Kunneth formula with coefficients over a Dedekind domain, which is used to prove a Kunneth like formula involves homologies over a Dedekind domain and generalizes similar results with integral or field coefficients. Over a principal ideal domain the formula for a free product is made more specific. Entrata in Redazione il 31 maggio 1978.     

Certain classes of sets in Banach spaces and a topological characterization of operators of type RN

We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let T be a continuous linear operator from a Banach space X to a Banach space Y. The following assertions are equivalent:

1. T is an operator of type RN;
2. for any Banach space Z, for any number p, p > 0, and any p-absolutely summing operator U:Z → X the operator TU is approximately p-Radonifying;
3. for any Banach space Z and any absolutely summing operator U:Z → X the operator TU is approximately 1-Radonifying.

We note that the implication I)?2), is apparently new even if the operator T is weakly compact.     

Semicontinuous Mappings into T.V.S. with Applications to Mixed Vector Variational-Like Inequalities

Let (X, Z) be the space of continuous linear mappings between topological vector spaces, where Z is Hausdorff and preordered by a closed convex cone C. In this paper, we introduce a notion of semicontinuity to any function from a topological space into X. A notion of semicontinuity is also introduced to any function from a topological space into (X, Z). These two notions of semicontinuity are related by the embedding of X into (X, Z). Their basic properties are given. As an application, we derive some existence results for the mixed vector variational-like inequality. This work was partially supported by grants from the National Science Council of the Republic of China.     

Countably Compact Paratopological Groups

By means of the theory of bispaces we show that a countably compact T₀ paratopological group (G, τ) is a topological group if and only if (G, τ ∨ τ^-1) is ω-bounded (here τ^-1 is the conjugate topology of τ). Our approach is premised on the fact that every paratopological countably compact paratopological group is a Baire space and on the notion of a 2-pseudocompact space. We also prove that every ω-bounded (respectively, topologically periodic) Baire paratopological group admits a weaker Hausdorff group topology. In particular, ω-bounded (respectively, topologically periodic) 2-pseudocompact (so, also countably compact) paratopological groups enjoy this property. Some topological properties turning countably compact topological semigroups into topological groups are presented and some open questions are posed.     

Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions

Let (E, τ) be a topological vector space and P a cone in E. We shall define a topology τ _P on E so that (E, τ _P ) is a normable topological vector space and P is a normal cone with normal constant M = 1. Then by using the norm, we shall give some results about common fixed points of two multifunctions on cone metric spaces.     

Factorization of mappings of topological spaces and homomorphisms of topological groups in accordance with weight and dimension ind

Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelöf number of a space X, respectively. We prove the following factorization theorems.

1. Let X and Y be Tychonoff spaces, φ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
2. Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and w(H)≤τ. Then there are a Hausdorff topological group G^* and continuous homomorphisms g: G→G^*, h: G^*→H so that π=h o g, G^*=g(G), w(G^*)≤τ andind G^*≤ind G. If nw(G)≤τ, then g is one-to-one.
3. For every continuous mapping φ: X→Y of a regular Lindelöf space X to a Tychonoff space Y one can find a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind ₀ Z≤ind ₀ X, whereind ₀ is the dimension function defined by V.V.Filippov with the help of G_δ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous result also holds for topological groups.
4. For each continuous homomorphism π: G→H of a Hausdorff Lindelöf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff group H there exist a Hausdorff group G^* and continuous homomorphisms g: G→G^*, h:G^*→H so that π=h o g, G^*=g(G), w(G^*)≤w(H),dimG^*≤dimG, andind G^*≤ind G. Bibliography: 25 titles.

     

On the compact-open and admissible topologies

It is known (see, for example, [H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Trans. Amer. Math. Soc. 336 (1) (1993) 101-119; M. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145; D.N. Georgiou, S.D. Iliadis, F. Mynard, Function space topologies, in: Open Problems in Topology 2, Elsevier, 2007, pp. 15-23]) that the intersection of all admissible topologies on the set C(Y,Z) of all continuous maps of an arbitrary space Y into an arbitrary space Z, is always the greatest splitting topology (which in general is not admissible). The following, interesting in our opinion, problem is arised: when a given splitting topology (for example, the compact-open topology, the Isbell topology, and the greatest splitting topology) is the intersection of k admissible topologies, where k is a finite number. Of course, in this case this splitting topology will be the greatest splitting.In the case, where a given splitting topology is admissible the above number k is equal to one. For example, if Y is a locally compact Hausdorff space, then k=1 for the compact-open topology (see [R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429-432; R. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480-495; R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31]). Also, if Y is a corecompact space, then k=1 for the Isbell topology (see [P. Lambrinos, B.K. Papadopoulos, The (strong) Isbell topology and (weakly) continuous lattices, in: Continuous Lattices and Applications, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 191-211; F. Schwarz, S. Weck, Scott topology, Isbell topology, and continuous convergence, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 251-271]).In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] a non-locally compact completely regular space Y is constructed such that the compact-open topology on C(Y,S), where S is the Sierpinski space, coincides with the greatest splitting topology (which is not admissible). This fact is proved by the construction of two admissible topologies on C(Y,S) whose intersection is the compact-open topology, that is k=2.In the present paper improving the method of [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] we construct some other non-locally compact spaces Y such that the compact-open topology on C(Y,S) is the intersection of two admissible topologies. Also, we give some concrete problems concerning the above arised general problem.