By a frechet space we mean a hausdorff, metrizable, complete locally convex space. Any normed vector space can be made into a metric space in a natural way. Here an order is compactly generated if it can be recovered from its speci cation over compact sets, that is if, so to speak, the information required to determine the order is local in nature. Frecheturyson space, sequential space, uniform space. G to be open if and only if 1 q u is open in mqx,p. A topological space x is exponentiablein the category top of topological spaces if for every space y there is a topology on the set cx,y of continuous maps from x to y such that for any space a the association f f. It is shown that any continuous action of a compactly generated topological group on a compact hausdorff space with vanishing topological entropy is amenable. Compactly generated group, a topological group which is algebraically generated by one of its compact subsets. Received by the editors may 5, 1997 and, in revised form, august 17, 2001 and october 17, 2001. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. Firstcountable implies compactly generated topospaces. Cones structures over topological manifolds determine orders which are of this type. E will always denote the topological dual of a locally convex space e. This paper is devoted to socalled compactly convex sets in linear topological spaces.
By weakening the approach t ised in borsuk in 2, considering only the compact subsets of the metric. This article gives the statement and possibly, proof, of an implication relation between two topological space properties. In this paper, we mainly discuss compactly generated recti able spaces or paratopological groups. A compactly generated hausdorff space is a topological space that is both a compactly generated space and a hausdorff space. The study of top and of properties of topological spaces using the techniques of. Specifically, a topological space x is compactly generated if it satisfies the following condition. The category top of compactly generated topological spaces is complete, cocomplete and cartesian closed more details for this kind of topological spaces in bro88, may99, the appendix of lew78 and also the preliminaries of gau03a. The term convenient category of topological spaces is used e. Xy, f is continuous if and only if f is k continuous. Pestov 1 mathematical notes of the academy of sciences of the ussr volume 40, pages 880 882 1986 cite this article. This leads us to the definition of a topological space. Introduction we answer in the negative the question posed in 7 and in 1, problem 5. Every cwcomplex is a compactly generated topological space.
In his 1967 paper a convenient category of topological spaces, norman steenrod introduced the category cgh of compactly generated hausdorff spaces as a good replacement of the category top topological spaces, in order to do homotopy theory the most important difference between cgh and top is that in cgh there is a functorial homeomorphism. If ctopis the category of topological spaces, a functor g. The next two propositions imply that the vast majority of spaces in common use are cgwh. Compact, compactly generated, topological group, metric space, metric, convergent sequence. That is, it states that every topological space satisfying the first topological space property i. A basis b for a topological space x is a set of open sets, called basic open sets, with the following properties. Ais a family of sets in cindexed by some index set a,then a o c. The original example of a convenient category of topological spaces is described at. Pdf topological entropy of continuous actions of compactly. Compactly generated recti able spaces or paratopological groups. It is shown that it is strictly broader than the class of hausdorff compactly generated spaces or kspaces and also that lspaces possess many of the nice properties of kspaces e. Intersections of compactly many open sets are open martn escard o. We introduce a notion of topological entropy for continuous actions of compactly generated topological groups on compact hausdorff spaces.
Comparing cartesian closed categories of core compactly. Fundamentals14 1 introduction 15 2 basic notions of pointset topology19 2. A recti able space or a paratopological group g is compactly generated if g k for some compact subset k of g. Compactly generated space, a topological space whose topology is coherent with the family of all compact subspaces. A topological space x is exponentiable in the category top of topological spaces if for every space y there is a topology on the set cx,y of continuous maps from x to y. Compactly generated spaces and quasispaces in topology. Any firstcountable space is compactly generated proof. Compactly generated quasitopological homotopy groups with. Topological entropy of continuous actions of compactly. This makes cghaus a particularly convenient category of topological spaces that is often used in place of top. Compactly generated quasitopological homotopy groups. A topological space x is exponentiable in the category top of topological spaces if for every space y there is a topology on the set cx,y of continuous maps from x to y such that for any space a the association f f is a bijection from the set of continuous maps ca. C is simply a group action on a topological space x.
The property of topological spaces of being firstcountable is stronger than the property of being compactly generated verbal statement. The category of compactly generated spaces will be denoted by cq. In recent years the category k of compactly generated spaces has received attention as being a very convenient setting for investigating various algebraic topological notions 7, 10, 11. In topology, a compactly generated space or k space is a topological space whose topology is coherent with the family of all compact subspaces. In mathematics, the category of topological spaces, often denoted top, is the category whose objects are topological spaces and whose morphisms are continuous maps. A locally convex space is said to be weakly compactly generated if there exists an increasing sequence of ge,ecompact subsets. Steenrod 67 for a category of topological spaces nice enough to address many of the needs of working topologists, notably including the condition of being a cartesian closed category. For the sequel, any topological space will be supposed to be compactly generated. Compactly generated topological groups springerlink.
A subset a xis called kclosed in xif for any compact hausdor space kand continuous map u. Pdf compactly convex sets in linear topological spaces. The concept of a realcompact topological space, although originally introduced and studied in general topology, has been also. X is called kclosed in x if for any compact hausdorff space k and continuous map u.
Why are compactly generated hausdorff spaces cartesian closed. Examples of topological spaces that fail to be compactly generated include the following. Feb, 2015 ous action of a compactly generated topological group on a compact hausdor. Nachbin 6 observed that, more generally, the intersection of compactly many. Then we call k k a norm and say that v,k k is a normed vector space. Let ibe the category whose objects and morphisms are determined by the following graph. We introduce a notion of topological entropy for continuous actions of compactly generated topological groups on compact hausdor. A topological space x is exponentiable in the category top of topological spaces if for every space y there is a topology on the set cx, y of continuous maps. Of course when we do this, we want these open sets to behave the way open sets should behave. The notions of compactness and hausdor separation for generalized enriched categories allow us, as classically done for the category top of topological spaces and continuous functions, to study compactly generated spaces and quasispaces in this setting. If b is a set satisfying these two properties, the topology generated by b is the set.
This chapter discusses the category cg of compactly generated spaces, which contains the category of cwcomplexes. Why the w in cgwh compactly generated weakly hausdorff. Compactly generated groups 305 component of mqx,p, and declare u. Compactly generated recti able spaces or paratopological. This disambiguation page lists articles associated with the title compactly generated. Chapter 9 the topology of metric spaces uci mathematics.
A topological space x is compactly generated if the following holds. Metricandtopologicalspaces university of cambridge. A topological space xis weakly hausdor wh if for every compact hausdor kand every continuous map u. Informally, 3 and 4 say, respectively, that cis closed under. Reminder about the compactly generated topological spaces. For any compactly generated topological space x and y, the space topx,yis the set of continuous maps from x to y equipped with the kelley. It is shown that any continuous action of a compactly generated topological group on a compact hausdor. Every locally compact space is compactly generated.
In topology, a compactly generated space or kspace is a topological space whose topology is. If v,k k is a normed vector space, then the condition du,v ku. A peano continuum is a compact locally path connected metric space. The colimit of this functor is the orbit space of the gaction on x.
Compactly generated spaces and quasispaces in topology willian ribeiro abstract. If q is a positive integer, a closed qcell is any space homeomorphic to 0,1q, and the qsphere sq is the quotient of 0,1q by identifying to a point the q1 dimensional boundary. I proved this claim for locally compact hausdorff spaces, where compact subsets are also closed. Any rstcountable topological space xin particular, any metrizable topological space is compactly generated. A subset a xis called kclosed in xif for any compact hausdor space kand continuous map. A recti able space or a paratopological group g is compactly generated if g k for some compact subset k. Appendix b compactly generated spaces sciencedirect. Top contains the full subcategory cghaus of compactly generated hausdorff spaces, which has the important property of being a cartesian closed category while still containing all of the typical spaces of interest. This article defines a property of topological spaces. As such, they are examples of nice categories of spaces a primary example is the category of compactly generated spaces. There is a set s of compact hausdorff spaces such that the. Boundedly generated topological spaces springerlink.
Two of these 2, 8 are compared with the compactly generated shape of this paper in 15, 17. It is equally as good a setting for topological algebraic investigations, since such objects as free modules. Xis a subset such that for all small compact hausdor spaces smapping to x, the preimage of v in sis closed. There exists a collection of compact subsets of, such that is open if and only if is open in for. The new class of boundedly generated topological spaces or lspaces is defined and studied by topological methods. Compactly generated spaces cwcomplexes are compactly generated spaces. One thing is that compact generation for weakly hausdorff spaces still takes the simple form that the space is the colimit of its compact subsets. Fucai lin1, 1 school of mathematics and statistics, minnan normal university, zhangzhou 363000, p.
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