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引用次数: 0
摘要
摘要给定M M和N N Hausdorff拓扑空间,我们研究了空间C0(M;N){C}^{0}\left(M;\hspace)上的拓扑{0.33em}N)连续映射f:M→ N f:M\到N。我们回顾了两种经典拓扑,“强”拓扑和“弱”拓扑。我们提出了一个“温和拓扑”的定义,它比“强”拓扑更粗糙,比“弱”拓扑更精细。我们比较了这三种拓扑的性质,特别是关于适当的连续映射f:M→ Nf:M\到N,以及当N=R N N={{\mathbb{R}}}^{N}时的仿射作用。为了定义“温和拓扑”,我们使用“分离函数”;这些“分离函数“与度量空间(M,d)\left(M,d)中通常的“距离函数d(x,y)d\left(x,y)”有些相似,但要求较弱。分离函数用于定义伪球,伪球是T2拓扑的全局基础。在一些额外的假设下,我们可以定义“集合分离函数”来证明拓扑是T6。此外,在进一步的假设下,我们将证明拓扑是可度量的。我们提供了分离函数使用的一些例子:一个是C 0(M;N){C}^{0}\left(M;\hspace上的温和拓扑的前面提到的情况{0.33em}N)。其他的例子是索根弗雷线和拓扑流形的拓扑。
Abstract 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.
期刊介绍:
Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed.
AGMS is devoted to the publication of results on these and related topics:
Geometric inequalities in metric spaces,
Geometric measure theory and variational problems in metric spaces,
Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density,
Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds.
Geometric control theory,
Curvature in metric and length spaces,
Geometric group theory,
Harmonic Analysis. Potential theory,
Mass transportation problems,
Quasiconformal and quasiregular mappings. Quasiconformal geometry,
PDEs associated to analytic and geometric problems in metric spaces.