Compactly generated space

In topology, a compactly generated space is a topological space 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 open in this definition. If X is coherent with any cover of compact subspaces in the above sense then it is, in fact, coherent with all compact subspaces.

A Hausdorff-compactly generated space or k-space is a topological space whose topology is coherent with the family of all compact Hausdorff subspaces. Sometimes in the literature a compactly generated space refers to a Hausdorff-compactly generated space. In these cases compactness is often explicitly redefined at the beginning to mean both compact and Hausdorff (and quasi-compact takes the meaning of compact). In this article we make a clear separation between compactly generated spaces and Hausdorff-compactly generated spaces, since the choice affects the statement of the associated theorems.

A compactly generated Hausdorff space is a compactly generated space that is also Hausdorff. This is not to be confused with a Hausdorff-compactly generated space which may or may not be Hausdorff. Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff or weakly Hausdorff. See the category of compactly generated weak Hausdorff spaces for the use in algebraic topology.

Motivation

Hausdorff-compactly generated spaces were originally called k-spaces, after the German word kompakt. They were studied by Hurewicz, 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 category of topological spaces. This fails to be a cartesian closed category, the usual cartesian product of identification maps is not always an identification map, and the usual product of CW-complexes need not be a CW-complex.^[1] 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 nLab on convenient categories of spaces.

The first suggestion (1962) to remedy this situation was to restrict oneself to the full subcategory of Hausdorff-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 generalize to the non-Hausdorff case;^[2] i.e. with compactly generated spaces. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.^[3]

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 Hausdorff-compactly generated (WHCG) spaces.

Examples and counterexamples

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-compactly generated, rather than just compactly generated, then its theorems often require an additional assumption of Hausdorffness somewhere.

Every Hausdorff-compactly generated space is compactly generated.
Every (Hausdorff) compact space is (Hausdorff-)compactly generated.
Every locally compact (Hausdorff) space is (Hausdorff-)compactly generated.
Every first-countable (Hausdorff) space is (Hausdorff-)compactly generated.
Topological manifolds are locally compact Hausdorff and therefore Hausdorff-compactly generated.
Metric spaces are first-countable Hausdorff and therefore Hausdorff-compactly generated Hausdorff.
Every CW complex is Hausdorff-compactly generated and Hausdorff.

Examples of topological spaces that fail to be compactly generated include the following:

The space $(\mathbb {R} \backslash \{1,{\frac {1}{2}},{\frac {1}{3}},\ldots \})\times (\mathbb {R} /\{1,2,3,\ldots \})$ , where the first factor uses the subspace topology, the second factor is the 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 on an infinite set $X$ , the induced topology has the property that every compact set is finite, and $X$ is not compactly generated.

Properties

Every locally closed subset of (Hausdorff)-compactly generated space is (Hausdorff)-compactly generated. A subset is locally closed if it is an intersection of an open subset and a closed subset.
A quotient of a (Hausdorff)-compactly generated space is (Hausdorff)-compactly generated.
A disjoint union of (Hausdorff)-compactly generated spaces is (Hausdorff)-compactly generated.
A wedge sum of (Hausdorff)-compactly generated spaces is (Hausdorff)-compactly generated.
The continuity of a map defined on a (Hausdorff-)compactly generated space X can be determined solely by looking at the compact (Hausdorff) subsets of X. Specifically, a function f : X → Y is continuous if and only if it is continuous when restricted to each compact (Hausdorff) subset K ⊆ X.
If $X$ is (Hausdorff-)compactly generated and $Y$ is locally compact (Hausdorff), then the product $X\times Y$ is (Hausdorff-)compactly generated.
If $X$ and $Y$ are two (Hausdorff-)compactly generated spaces, then $X\times Y$ may not be (Hausdorff-)compactly generated. Therefore, when working in categories of (Hausdorff-)compactly generated spaces it is necessary to define the product as (X × Y)_c, the k-ification of the product topology (see below).

K-ification

Given any topological space X we can define a possibly finer topology on X that is compactly generated, sometimes called the k-ification of the topology. Let {K_α} 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 subsets 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. We denote CGTop the full subcategory of Top with objects the compactly generated spaces, and CGHaus the full subcategory of CGTop with objects the Hausdorff spaces. The functor from Top to CGTop that takes X to X_c is right adjoint to the inclusion functor CGTop → Top.

The above discussion applies also to the Hausdorff-compactly generated spaces after replacing compact with compact Hausdorff, but with the following difference. To prove that the compact Hausdorff subsets of the k-ification are the same as in the original topology (and hence that the k-ification is Hausdorff-compactly generated) requires that the original topology is also k-Hausdorff. The following properties are equivalent:

Hausdorff-compactly generated k-Hausdorff
Hausdorff-compactly generated weak Hausdorff
Compactly generated k-Hausdorff

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.^[2] This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.

References

^ Hatcher, Allen (2001). Algebraic Topology (PDF). (See the Appendix)
^ ^a ^b Brown, Ronald (2006). Topology and Groupoids. Charleston, South Carolina: Booksurge. ISBN 1-4196-2722-8. (See section 5.9)
^ 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.

Overview

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

Other

Munkres, James (2000). Topology. Pearson Modern Classics for Advanced Mathematics Series (2nd ed.). Prentice-Hall, Inc. ISBN 9780134689517.
Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8.
Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.
J. Peter May, A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics ISBN 0-226-51183-9 (See Chapter 5.)
Strickland, Neil P. (2009). "The category of CGWH spaces" (PDF).

[hatcher-1] Hatcher, Allen (2001). Algebraic Topology (PDF). (See the Appendix)

[brown-2] Brown, Ronald (2006). Topology and Groupoids. Charleston, South Carolina: Booksurge. ISBN 1-4196-2722-8. (See section 5.9)

[3] 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.

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