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Ultrafilter-limit points in metric dynamical systems

Salvador García-FerreiraManuel Sanchis — 2007

Commentationes Mathematicae Universitatis Carolinae

Given a free ultrafilter p on and a space X , we say that x X is the p -limit point of a sequence ( x n ) n in X (in symbols, x = p - lim n x n ) if for every neighborhood V of x , { n : x n V } p . By using p -limit points from a suitable metric space, we characterize the selective ultrafilters on and the P -points of * = β ( ) . In this paper, we only consider dynamical systems ( X , f ) , where X is a compact metric space. For a free ultrafilter p on * , the function f p : X X is defined by f p ( x ) = p - lim n f n ( x ) for each x X . These functions are not continuous in general. For a...

Some remarks on the product of two C α -compact subsets

Salvador García-FerreiraManuel SanchisStephen W. Watson — 2000

Czechoslovak Mathematical Journal

For a cardinal α , we say that a subset B of a space X is C α -compact in X if for every continuous function f X α , f [ B ] is a compact subset of α . If B is a C -compact subset of a space X , then ρ ( B , X ) denotes the degree of C α -compactness of B in X . A space X is called α -pseudocompact if X is C α -compact into itself. For each cardinal α , we give an example of an α -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness”...

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