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Given M M and N N Hausdorff topological spaces, we study topologies on the space C 0 (M ; N) {C}^{0}\left(M;\hspace{0.33em}N) of continuous maps f : M → N f:M\to N. We review two classical topologies, the "strong" and the "weak" topology. We propose a definition of "mild topology" that is coarser than the "strong" and finer than the "weak" topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f : M → N f:M\to N , and affine actions when N = R n N={{\mathbb{R}}}^{n}. To define the "mild topology" we use "separation functions;" these "separation functions" are somewhat similar to the usual "distance function d (x , y) d\left(x,y) " in metric spaces (M , d) \left(M,d) , but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define "set separation functions" that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions: one is the aforementioned case of the mild topology on C 0 (M ; N) {C}^{0}\left(M;\hspace{0.33em}N). Other examples are the Sorgenfrey line and the topology of topological manifolds. [ABSTRACT FROM AUTHOR]

The purpose of this investigation is to define the weak topology on a specific topological spaces (modular spaces) this topology generated by the modular dual space X* i.e. it is generated by class of mappings F = {fα: X → Yα:fα linear and continuous} when F =X*, we denote it σ(XM,X*) and induced characterization to the structure of this topology. Afterwards, we proof it is hausdorff space finally we discuss the property of convergence inσ(XM,X*). [ABSTRACT FROM AUTHOR]

A topology τ on a nonempty set X is called a clopen topology provided each member of τ is both open and closed. Given a function f from X to Y, the operator E 7→ f-1( f (E)) is a closure operator on the power set of X whose fixed points are closed subsets corresponding to a clopen topology on X. Conversely, for each clopen topology τ on X, we produce a function f with domain X such that τ = {E ⊆ X : E = f-1( f (E))}. We characterize the clopen topologies on X as those that are weak topologies determined by a surjective function with values in some discrete topological space. Paralleling this result, we show that a topology admits a clopen base if and only if it is a weak topology determined by a family of functions with values in discrete spaces. [ABSTRACT FROM AUTHOR]

In this paper we prove that the eigenvalues of a vibrating beam have a strongly continuous dependence on the elastic destructive force, i.e., the eigenvalues, as nonlinear functionals of the elastic destructive force, are continuous in the elastic destructive force with respect to the weak topologies in the Lebesgue spaces L p . In virtue of the minimax characterization for eigenvalues, we prove first the continuity of the lowest eigenvalue and then all the eigenvalues by the induction principle. [ABSTRACT FROM AUTHOR]

In this paper we will prove that the eigenvalues of nonhomogeneous hinged vibrating rods have a strongly continuous dependence on weights, i.e., as nonlinear functionals of weights, eigenvalues are continuous in weights with respect to the weak topologies in the Lebesgue spaces L p . [ABSTRACT FROM AUTHOR]

We describe the topology of any cosmic space and any ℵ 0 -space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ ( E , E ′ ) are ℵ 0 -spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) ℓ 1 and satisfying the Heinrich density condition we prove that ( E , σ ( E , E ′ ) ) is an ℵ 0 -space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain ℓ 1 , then ( E , σ ( E , E ′ ) ) is an ℵ 0 -space if and only if E ′ is separable. The last part of the paper studies the question: Which spaces ( E , σ ( E , E ′ ) ) are ℵ 0 -spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a ( DF )-space whose strong dual E ′ is separable, then ( E , σ ( E , E ′ ) ) is an ℵ 0 -space. Supplementing an old result of Corson we show that, for a Čech-complete Lindelöf space X the following are equivalent: (a) X is Polish, (b) C c ( X ) is cosmic in the weak topology, (c) the weak ⁎ -dual of C c ( X ) is an ℵ 0 -space. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with existence results for a nonlinear Hammerstein integral equation inL1spaces. Making use of the De Blasi measure of weak noncompactness, we establish variants of some Krasnosel'skii type theorem involving the weak topology of Banach spaces. [ABSTRACT FROM AUTHOR]

We present a convergence theorem for the Henstock-Kurzweil integral of functions taking values in a locally convex topological vector space, which is sequentially complete with respect to its weak topology. [ABSTRACT FROM AUTHOR]

Abstract: Given a 1-periodic real potential . We use to denote the smallest 1-periodic eigenvalue of the Hill''s equation . Let be the ball centered at 0 of radius r in the space . It is trivial that for all . Based on continuity of in q with the weak topology and continuous differentiability of in q with the norm , we will apply scaling technique, variational approach to extremal values in balls, singular integrals and the limiting approach as to obtain (i) is bounded for q in any bounded set of , and (ii) the minimal value is simply , where , . The extremal values of the smallest Neumann eigenvalues for potentials in balls are also found explicitly. [Copyright &y& Elsevier]

Abstract: We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space , we also consider the pcc-property of the topology of pointwise convergence, and we note that the space may be pcc even when fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example: [–] There exists a non-metrizable scattered compact Hausdorff space K with weakly pcc. [Copyright &y& Elsevier]

Abstract: The replicator equation model for the evolution of individual behaviors in a single species with a multi-dimensional continuous trait space is developed as a dynamics on the set of probability measures. Stability of monomorphisms in this model using the weak topology is compared to more traditional methods of adaptive dynamics. For quadratic fitness functions and initial normal trait distributions, it is shown that the multi-dimensional continuously stable strategy (CSS) of adaptive dynamics is often relevant for predicting stability of the measure-theoretic model but may be too strong in general. For general fitness functions and trait distributions, the CSS is related to dominance solvability which can be used to characterize local stability for a large class of trait distributions that have no gaps in their supports whereas the stronger neighborhood invader strategy (NIS) concept is needed if the supports are arbitrary. [Copyright &y& Elsevier]