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).     

Topologies on function spaces

In the present paper we introduce notions of A-splitting and A-jointly continuous topology on the set C(Y,Z) of all continuous maps of a topological space Y into a topological space Z, where A is any family of spaces. These notions satisfy the basic properties of splitting and jointly continuous topologies on C(Y,Z). In particular, for every A, the greatest A-splitting topology on C(Y,Z) (denoted by τ(A) always exists. We indicate some families A of spaces for which the topology τ(A) coincides with the greatest splitting topology on C(X,Y). We give a notion of equivalent families of spaces and try to find a “simple” family which is equivalent to a given family. In particular, we prove that every family is equivalent to a family consisting of one space, and the family of all spaces is equivalent to a family of all T₁-spaces containing at most one nonisolated point. We compare the topologies τ({X}) for distinct compact metrizable spaces X and give some examples. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 82–97. Translated by A. A. Ivanov.     

Pairings of function spaces

Pairings and copairings of topological spaces induce pairings of function spaces. These induced pairings of function spaces are studied. For this purpose, the C-open topology of function spaces is studied for subcategories C of Top. It is shown that the C-open topology enjoys good properties for homotopy theory. Making use of the C-open topology, theory of induced pairings is established and fundamental results on pairings which deduce various commutativity properties of elements in homotopy set are extended to function spaces.     

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.     

Unique Nearness Structures

We investigate which symmetric topological spaces have precisely one compatible nearness structure. We show how this situation exists for T ₁ topological spaces and for T ₂ topological spaces. The compatible nearness structures on a symmetric topological space form a sort of “interval” of structures with the topology itself as the topmost.     

CONTINUITY OF THE ONE-SIDED BEST UNIFORM APPROXIMATION OPERATOR

Abstract

The purpose of this paper is to relate the continuity and selection properties of the one-sided best uniform approximation operator to similar properties of the metric projection. Let M be a closed subspace of C(T) which contains constants. Then the one-sided best uniform approximation operator is Hausdorff continuous (resp. Lipschitz continuous) on C(T) if and only if the metric projection P_M is Haudorff continuous (resp. Lipschitz continuous) on C(T). Also, the metric projection P_M admits a continuous (resp. Lipschitz continuous) selection if and only if the one-sided best uniform approximation operator admits a continuous (resp. Lipschitz continuous) selection.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

Properties of topologies of information retrieval systems

This paper studies topological properties of different topologies that are possible on the space of documents as they are induced by queries in a query space together with a similarity function between queries and documents. The main topologies studied here are the retrieval topology (introduced by Everett and Cater) and the similarity topology (introduced by Egghe and Rousseau).The studied properties are the separation properties T₀, T₁, and T₂ (Hausdorff), proximity and connectedness. Full characterizations are given for the diverse topologies to be T₀, T₁, or T₂. It is shown that the retrieval topology is not necessarily a proximity space, while the similarity topology and the pseudo-metric topology always are proximity spaces. A characterization of connectedness in terms of the Boolean NOT-operator is given, hereby showing the intimate relationship between IR and topology.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

ON CERTAIN INITIAL COMPLETIONS IN FUZZY TOPOLOGY

Abstract

Following a lead given by I.W. Alderton, it is shown that the MacNeille completion and the universal initial completion coincide for the categories of zero-dimensional fuzzy T₀-topological spaces, T₀-fuzzy closure spaces, 2T ₀-fuzzy bitopological spaces, and T ₁-fuzzy topological spaces and that these turn out to be respectively the categories of zero-dimensional fuzzy topological spaces, fuzzy closure spaces, fussy bitopological spaces, and fuzzy R ₀ topological spaces.     

On Q-sobriety

The study of fixed-basis variety-based topology was initiated by S.A. Solovyov (in 2008), which, among other things, generalizes fuzzy topology. We extend within this framework, an earlier result due to Srivastava et al. (in 1998), which showed that the category of sober fuzzy topological spaces is the epireflective hull of the fuzzy Sierpinski space in the category of T₀-fuzzy topological spaces.     

Special Uniform Approximations of Continuous Vector-Valued Functions. Part I: Special Approximations in CX(T)

In this paper, we give special uniform approximations of functions u from the spaces C_X(T) and C_∞(T,X), with elements of the tensor products C_Γ(T)X, respectively C₀(T,Γ)X, for a topological space T and a Γ-locally convex space X. We call an approximation special, if satisfies additional constraints, namely supp vu⁻¹(X\{0}) and (T) co(u(T)) (resp. co(u(T){0})). In Section 3, we give three distinct applications, which are due exactly to these constraints: a density result with respect to the inductive limit topology, a Tietze–Dugundji's type extension new theorem and a proof of Schauder–Tihonov's fixed point theorem.     

Polyhedral norms on non-separable Banach spaces

We prove the existence of equivalent polyhedral norms on a number of classes of non-separable spaces, the majority of which being of the form C(K). In particular, we obtain a complete characterization of those trees T, such that C₀(T) admits an equivalent polyhedral norm.     

Algebraic cocycles on normal, quasi-projective varieties

This paper extends to quasi-projective varieties earlier work by the author and H. Blaine Lawson concerning spaces of algebraic cocycles on projective varieties. The topological monoid C_r(Y) (U) of effective cocycles on a normal, quasi-projective variety U with values in a projective variety Y consists of algebraic cycles on U×Y equi-dimensional of relative dimension r over U. A careful choice of topology enables the establishment of various good properties: the definition is essentially algebraic, the group completion Z_r (Y) (U) has 'sensible' homotopy groups, the construction is contravariant with respect to U, convariant with respect to Y, and there is a natural 'quality map" to the topological group of cycles on U×Y. The fundamental theorem presented here is the extension of Friedlander-Lawson duality to this context: the duality map Z_r (Y) (U) to Z_{_r+m} (U × Y) is a homotopy equivalence provided that both U and Y are smooth (where m=dim U). Various application are given, especially the determination of the homotopy types of certain topological groups of algeb raic morphisms.     

An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff.     

Chebyshev diagrams for two-bridge knots

We show that every two-bridge knot K of crossing number N admits a polynomial parametrization x = T ₃(t), y = T _b (t), z = C(t), where T _k (t) are the Chebyshev polynomials and b + deg C = 3N. If C(t) = T _c (t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for a ≤ 3. Most results are derived from continued fractions and their matrix representations.     

Multiplikatorsysteme der symplektischen Thetagruppe

Let Γ_θ be the subgroup of Siegel modular groupSp(n, ?) consisting of all matrices \(M = \left( {\begin{array}{*{20}c} {A B} \\ {C D} \\ \end{array} } \right)\) , such that the diagonal elements ofA ^t C andB ^t D are even. A multiplier system of weightr(∈?) is a system of complex numbers ν (M)≠0,M∈Γ_θ, such thatJ (M, Z)=ν(M) det(CZ+D) ^r is an automorphy factor (that isJ (M N, Z)=J (M, N Z) J (N, Z) forM, N ∈Sp(n,?) and $$Z \in S_n = \left\{ {Z = X + i Y \in M^{(n,n)} (\mathbb{C}); X = X^t , Y = Y^t > 0} \right\})$$ . We show that in casen≥2 such a multiplier system exists if and only if 2r∈?. A corollary of this fact is the following. From the cohomology theory of Siegel modular group we derive that in casen≥8 any Γ_θ-invariant divisor is the exact zero divisor of a modular form for Γ_θ. Therefore the zero divisor of classical theta function \(\theta (Z) = \sum\limits_{g \in \mathbb{Z}^n } {e^{\pi iZ[g]} } \) , a modular form of weight 1/2 is irreducible. In the second part of this paper we calculate the commutator factor group of Γ _{n, θ} forn≥2.     

Generalized Linear Systems on Curves and Their Weierstrass Points

Let C be a projective Gorenstein curve over an algebraically closed field of characteristic 0. A generalized linear system on C is a pair (?, ε) consisting of a torsion-free, rank-1 sheaf ? on C, and a map of vector spaces ε: V → Γ(C, ?). If the system is nondegenerate on every irreducible component of C, we associate to it a 0-cycle W, its Weierstrass cycle. Then we show that for each one-parameter family of curves C _t degenerating to C, and each family of linear systems (? _t , ε _t ) along C _t , with ? _t invertible, degenerating to (?, ε), the corresponding Weierstrass divisors degenerate to a subscheme whose associated 0-cycle is W. We show that the limit subscheme contains always an “intrinsic” subscheme, canonically associated to (?, ε), but the limit itself depends on the family ? _t .