Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ?) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

The strict topology in non-Archimedean vector-valued function spaces

Let F be a non-trivial complete non-Archimedean valued field. We study the strict topology β₀ on the space C_b(X,E) of all bounded continuous functions from a topological space X to a non-Archimedean F-locally convex space E over F. We also show that the dual of the space (C_b(X,E), β_o) is a certain space of E′-valued measures and we give a characterization of the equicontinuous subsets of this dual space.     

The Vietoris topology on rectifiable spaces

A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism φ:G×G→G×G and an element e∈G such that π ₁°φ=π ₁ and for every x∈G we have φ(x,x)=(x,e), where π ₁:G×G→G is the projection to the first coordinate. Let G be a rectifiable space and C(G) be the family of all non-empty compact subsets of G. In this paper, we study the Vietoris topology on C(G), and show that if G is a locally compact rectifiable space, then (C(G),?) together with the Vietoris topology is a topological semi-right loop.     

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.     

Sequential properties of function spaces with the compact-open topology

The main results of the paper are:

(1)

If X is metrizable but not locally compact topological space, then Ck(X) contains a closed copy of S₂, and hence does not have the property AP;

(2)

For any zero-dimensional Polish X, the space Ck(X,2) is sequential if and only if X is either locally compact or the derived set X^′ is compact; and

(3)

All spaces of the form Ck(X,2), where X is a non-locally compact Polish space whose derived set is compact, are homeomorphic, and have the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n+1)st.

     

String topology on Gorenstein spaces

The purpose of this paper is to describe a general and simple setting for defining (g, p + q)-string operations on a Poincaré duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincaré duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each (g, p + q)-string operation as well as it leads to explicit computations.     

The topology of spaces of morse functions on surfaces

The topology of the space F = F(M) of Morse functions on a compact smooth orientable two-dimensional surface M is studied.     

The topology of spaces of coprime polynomials

This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

A strict topology on Orlicz spaces

Let φ be a Young function, Ω be a locally compact space, and μ be a positive Radon measure on Ω. We consider a strict topology (in the sense of Sentilles‐Taylor) on the Orlicz function space and investigate various properties of this locally convex topology. We also study the Orlicz space of a locally compact group G with a left Haar measure under the strict topology and certain other natural locally convex topologies. Finally we present some results on various continuity properties of convolution operators on under the topology and other natural ones.     

Higher string topology on general spaces

In this paper, I give a generalized analogue of the string topologyresults of Chas and Sullivan, and of Cohen and Jones. For afinite simplicial complex X and k 1, I construct a spectrumMaps(S^k, X)^S(X), which is obtained by taking a generalizationof the Spivak bundle on X (which however is not a stable spherebundle unless X is a Poincaré space), pulling back toMaps(S^k, X) and quotienting out the section at infinity. I showthat the corresponding chain complex is naturally homotopy equivalentto an algebra over the (k + 1)-dimensional unframed little diskoperad C_{k + 1}. I also prove a conjecture of Kontsevich, whichstates that the Quillen cohomology of a based C_k-algebra (inthe category of chain complexes) is equivalent to a shift ofits Hochschild cohomology, as well as prove that the operadC_*C_k is Koszul-dual to itself up to a shift in the derived category.This gives one a natural notion of (derived) Koszul dual C_*C_k-algebras.I show that the cochain complex of X and the chain complex of^k X are Koszul dual to each other as C_*C_k-algebras, and thatthe chain complex of Maps(S^k, X)^S(X) is naturally equivalentto their (equivalent) Hochschild cohomology in the categoryof C_* C_k-algebras. 2000 Mathematics Subject Classification 55P48(primary), 16E40, 55N45, 18D50 (secondary).     

The finite topology and variational calculus on nontopological vector spaces

Certain aspects of the calculus of variations are presented in the setting of nontopological vector spaces, and the results are shown to have certain advantages in the investigation of various optimization problems of economics that seem more directly accessible by these techniques than by the maximum principle of optimal control theory.     

Cosmic spaces which are not μ-spaces among function spaces with the topology of pointwise convergence

A space is called a μ-space if it can be embedded in a countable product of paracompact F_σ-metrizable spaces. The following are shown:(1) For a Tychonoff space X, if C_p(X,R) is a μ-space, then X is a countable union of compact metrizable subspaces.(2) For a zero-dimensional space X, C_p(X,2) is a μ-space if and only if X is a countable union of compact metrizable subspaces.In particular, let P be the space of irrational numbers. Then C_p(P,2) is a cosmic space (i.e., a space with a countable network) which is not a μ-space.     

Advanced topology on the multiscale sequence spaces

We pursue the study of the multiscale spaces S^ν introduced by Jaffard in the context of multifractal analysis. We give the necessary and sufficient condition for S^ν to be locally p-convex, and exhibit a sequence of p-norms that defines its natural topology. The strong topological dual of S^ν is identified to another sequence space depending on ν, endowed with an inductive limit topology. As a particular case, we describe the dual of a countable intersection of Besov spaces.     

Metric spaces in synthetic topology

We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree.     

The topology of spaces of phylogenetic trees with symmetry

Natural Dowling analogues of the complex of phylogenetic trees are studied. Using discrete Morse theory, we find their homotopy types. In the process, the homotopy types of certain subposets of Dowling lattices are determined.     

The topology of moduli spaces of free group representations

For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the -character variety of a rank 2 free group is homotopic to an 8 sphere and the -character variety of a rank 3 free group is homotopic to a 6 sphere.