2024 Compactly generated spaces

Compactly generated spaces

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August undefined, 2024

WebFor any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g., that of compactly generated spaces ), there are natural (basepoint preserving) homeomorphisms However, for the naive category of pointed spaces, this fails, as shown by the counterexample and found by Dieter Puppe. [1] WebOct 20, 2012 · As regards the second question, I have the feeling that functional analysts and topologists use the term compactly generated with different meanings. For the former, a locally convex space (in particular, a Banach space) is compactly generated if it contains a compact subset whose span is dense.

Unraveling the various definitions of $k$-space or compactly generated ...

WebAug 22, 2024 · Appendix A - Compactly generated spaces Published online by Cambridge University Press: 22 August 2024 Stefan Schwede Chapter Get access Share Cite Summary A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. … WebWhen working in the category of compactly generated spaces, it is common to modify this definition by restricting to the subbase formed from those K that are the image of a … \u0027sdeath t3

Why the "W" in CGWH (compactly generated weakly …

Web1) In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological … WebTo ﬁx notations, let Udenote the category of compactly generated Hausdorﬀ spaces and continuous maps, and let Tdenote the category of based compactly generated Hausdorﬀ spaces and based maps. Base-points will always be denoted by ∗and will by required to be non-degenerate, in the sense that (X,∗) is an NDR-pair for X∈T. Products ... Webde nes a functor from pointed spaces to pointed spaces. On maps, a map f: X 0!Xis sent to map X ! Xin the obvious way: post-composing a loop into X0by f. (a)Let X;Y;Zbe compactly generated spaces. Prove there is a homeomorphism of spaces Maps (X^Y;Z) ˘=Maps (X;Maps (Y;Z)): (b)Let Xand Zbe compactly generated. Show there is an isomorphism … \u0027sdeath t2

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WebThe notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $$\\textsf {Top}$$ Top of topological spaces and continuous functions, to study compactly generated spaces and quasi-spaces in this setting. Moreover, for a class $$\\mathcal {C}$$ C of objects we generalize the notion of … In topology, a topological space $${\displaystyle X}$$ is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not … See more General framework for the definitions Let $${\displaystyle (X,T)}$$ be a topological space, where $${\displaystyle T}$$ is the topology, that is, the collection of all open sets in $${\displaystyle X.}$$ There are multiple … See more • In a compactly generated space, every closed set is compactly generated. The same does not hold for open sets. For example, if $${\displaystyle X}$$ denotes the Arens-Fort space, the one-point compactification of $${\displaystyle X}$$ is compact, hence … See more • Compact-open topology • Countably generated space – topological space in which the topology is determined by its countable subsets • CW complex – Type of topological space See more Compactly generated spaces were originally called k-spaces, after the German word kompakt. They were studied by See more Most topological spaces commonly studied in mathematics are (Hausdorff-)compactly generated. In the following the bracketed (Hausdorff) properties and (Hausdorff-) prefixes are meant to be applied together. Generally, if the space is Hausdorff … See more Given any topological space $${\displaystyle X}$$ we can define a possibly finer topology on $${\displaystyle X}$$ that … See more • Compactly generated spaces - contains an excellent catalog of properties and constructions with compactly generated spaces • Compactly generated topological space at the nLab • Convenient category of topological spaces at the nLab See more \u0027sdeath t4WebAug 28, 2004 · It is well known that, although the category of topological spaces is not Cartesian closed, it possesses many Cartesian closed full subcategories, e.g.: (i) … \u0027sdeath t0

"WebOct 18, 2010 · In this chapter we study weakly compact operators and the related class of Banach spaces that are generated by weakly compact sets (i.e., weakly compactly generated spaces, in short WCG spaces). We focus on their decomposition properties, renormings, and on the topological properties of their dual spaces. " - Compactly generated spaces

Unraveling the various definitions of $k$-space or compactly generated ...

Why the "W" in CGWH (compactly generated weakly …

Compactly generated spaces

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