compactly generated space

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In
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

, a compactly generated space (or k-space) is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subspace ''A'' is closed in ''X'' if and only if ''A'' ∩ ''K'' is closed in ''K'' for all compact subspaces ''K'' ⊆ ''X''. Equivalently, one can replace ''closed'' with ''
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'' in this definition. If ''X'' is coherent with any
cover Cover or covers may refer to: Packaging, science and technology * A covering, usually - but not necessarily - one that completely closes the object ** Cover (philately), generic term for envelope or package ** Housing (engineering), an exterior ...
of compact subspaces in the above sense then it is, in fact, coherent with all compact subspaces. A compactly generated Hausdorff space is a compactly generated space that is also . Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff or weakly Hausdorff.

# Motivation

Compactly generated spaces were originally called k-spaces, after the German word ''kompakt''. They were studied by
Hurewicz Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician. Early life and education Witold Hurewicz was born in Łódź, at the time one of the main Polish industrial hubs with economy focused on the textile industry. His f ...

, and can be found in General Topology by Kelley, Topology by Dugundji, Rational Homotopy Theory by Félix, Halperin, and Thomas. The motivation for their deeper study came in the 1960s from well known deficiencies of the usual
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. This fails to be a
cartesian closed category In category theory, a category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories' ...
, the usual
cartesian product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of
identification map as the quotient space of a Disk (mathematics), disk, by ''gluing'' together to a single point the points (in blue) of the boundary of the disk. In topology and related areas of mathematics, the quotient space of a topological space under a given e ...
s is not always an identification map, and the usual product of
CW-complexA CW complex is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better c ...
es need not be a CW-complex. ''(See the Appendix)'' By contrast, the category of simplicial sets had many convenient properties, including being cartesian closed. The history of the study of repairing this situation is given in the article on the ''n''Lab o
convenient categories of spaces
The first suggestion (1962) to remedy this situation was to restrict oneself to the full subcategory of compactly generated Hausdorff spaces, which is in fact cartesian closed. These ideas extend on the de Vries duality theorem. A definition of the exponential object is given below. Another suggestion (1964) was to consider the usual Hausdorff spaces but use functions continuous on compact subsets. These ideas can be generalised to the non-Hausdorff case. ''(See section 5.9)'' This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.P. I. Booth and J. Tillotson,
Monoidal closed, Cartesian closed and convenient categories of topological spaces
, ''Pacific Journal of Mathematics'', 88 (1980) pp.33-53.
In modern-day algebraic topology, this property is mostly commonly coupled with the weak Hausdorff property, so that one works in the category of weak Hausdorff compactly generated (WHCG) spaces.

# Examples and counterexamples

Most topological spaces commonly studied in mathematics are compactly generated. *Every Hausdorff compact space is compactly generated. *Every Hausdorff locally compact space is compactly generated. *Every first-countable space is compactly generated. *Topological manifolds are locally compact Hausdorff and therefore compactly generated Hausdorff. *Metric spaces are first-countable and therefore compactly generated Hausdorff. *Every CW complex is compactly generated Hausdorff. Examples of topological spaces that fail to be compactly generated include the following. * The space $\left(\mathbb R \backslash \\right) \times \left(\mathbb R/\\right)$, where the first factor uses the subspace topology, the second factor is the quotient space (topology), quotient space of R where all natural numbers are identified with a single point, and the product uses the product topology. * If $\mathcal F$ is a non-principal Ultrafilter (set theory), ultrafilter on an infinite set $X$, the induced topology has the property that every compact set is finite, and $X$ is not compactly generated.

# Properties

We denote CGTop the full subcategory of category of topological spaces, Top with objects the compactly generated spaces, and CGHaus the full subcategory of CGTop with objects the Hausdorff spaces. Given any topological space ''X'' we can define a (possibly) finer topology on ''X'' that is compactly generated. Let denote the family of compact subsets of ''X''. We define the new topology on ''X'' by declaring a subset ''A'' to be closed if and only if ''A'' ∩ ''K''α is closed in ''K''α for each α. Denote this new space by ''X''c. One can show that the compact subsets of ''X''c and ''X'' coincide, and the induced topologies on compact sets are the same. It follows that ''X''c is compactly generated. If ''X'' was compactly generated to start with then ''X''c = ''X'' otherwise the topology on ''X''c is strictly finer than ''X'' (i.e. there are more open sets). This construction is functorial. The functor from Top to CGTop that takes ''X'' to ''X''c is adjoint functors, right adjoint to the Subcategory#Formal definition, inclusion functor CGTop → Top. The continuity (topology), continuity of a map defined on a compactly generated space ''X'' can be determined solely by looking at the compact subsets of ''X''. Specifically, a function ''f'' : ''X'' → ''Y'' is continuous if and only if it is continuous when restricted to each compact subset ''K'' ⊆ ''X''. If ''X'' and ''Y'' are two compactly generated spaces the product topology, product ''X'' × ''Y'' may not be compactly generated (it will be if at least one of the factors is locally compact). Therefore when working in categories of compactly generated spaces it is necessary to define the product as (''X'' × ''Y'')c. The exponential object in CGHaus is given by (''Y''''X'')c where ''Y''''X'' is the space of continuous maps from ''X'' to ''Y'' with the compact-open topology. These ideas can be generalised to the non-Hausdorff case. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.

* Compact-open topology * Countably generated space * CW complex * Finitely generated space * K-space (functional analysis) * Weak Hausdorff space

# References

## Overview

Compactly generated spaces
- contains an excellent catalog of properties and constructions with compactly generated spaces * *

## Other

* * * J. Peter May,
A Concise Course in Algebraic Topology
', (1999) Chicago Lectures in Mathematics ''(See Chapter 5.)'' *{{cite web , last = Strickland , first = Neil P. , author-link = Neil Strickland , title = The category of CGWH spaces , year = 2009 , url = http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf General topology Homotopy theory