The categories of flows of Set and Top

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• 作者：Othman Echi ^{othechi@yahoo.com}
• 关键词：54H20 ; 54B30 ; 18A40
• 刊名：Topology and its Applications
• 出版年：2012
• 期刊代码：60_01668641
• 类别：mt
• 出版时间：1 June, 2012
• 卷：159
• 期：9
• 页码：2357-2366
• 文件大小：199 K

摘要

Following John Kennison, a flow (or discrete dynamical system) in a category C is a couple , where X is an object of C and is a morphism, called the iterator. If and are flows in C, then is a morphism of flows from to if . We let denote the resulting category of flows in C.

This paper deals with and , where Set and Top denote respectively the categories of sets and topological spaces.

By a Gottschalk flow, we mean a flow in Top satisfying the following conditions:

(i)

If is any almost periodic point of f, then the closure is a minimal set of f;

(ii)

All points in any minimal set of f are almost periodic points.

As proven by Gottschalk, if X is a compact Hausdorff space and is a continuous function, then is a Gottschalk flow.

In this paper, we prove that for any flow of Set, there is a topology on X for which is a Gottschalk flow in Top. This, actually, defines a covariant functor from into .

The main result of this paper provides a characterization of spaces in the image of the functor in order-theoretical terms.

Some categorical properties of and are also given.