Compactly generated space

A subset ${\displaystyle A\subseteq X}$ of a topological space ${\displaystyle X}$ is called k-closed (respectively, k-open) if for every compact subspace ${\displaystyle K\subseteq X,}$ ${\displaystyle A\cap K}$ is a closed (respectively, open) subset of ${\displaystyle K}$ (when ${\displaystyle K}$ is endowed with the subspace topology induced on it by ${\displaystyle X}$[note 1]). In all topological spaces, every closed subset is always k-closed and every open subset is always k-open,[1] although the converses are not guaranteed in general. Those spaces in which the converses are guaranteed are called compactly generated. Explicitly, a compactly generated space or k-space[1] is a topological space in which every k-closed subset is closed, or equivalently, it is a space in which every k-open subset is open.

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.

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.[2] 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;[3] i.e. with compactly generated spaces. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.[4]

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.

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.[1]
• Every first-countable (Hausdorff) space is (Hausdorff-)compactly generated.[1]
• 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.
• Every sequential space is a quotient of a metric space and therefore Hausdorff-compactly generated.

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

• The product space ${\displaystyle (\mathbb {R} \setminus \{1,{\tfrac {1}{2}},{\tfrac {1}{3}},\ldots \})\times (\mathbb {R} /\{1,2,3,\ldots \})}$ endowed with the product topology, where the first factor uses the subspace topology while the second factor is the quotient space of ${\displaystyle \mathbb {R} }$ where all natural numbers are identified with a single point.
• If ${\displaystyle {\mathcal {F}}}$ is a non-principal ultrafilter on an infinite set ${\displaystyle X,}$ the induced topology has the property that every compact set is finite, and ${\displaystyle X}$ is not compactly generated.

A subspace of a compactly generated space is not necessarily compact generated.[5] Although ${\displaystyle \mathbb {R} }$ is compactly generated, the product of uncountably many copies of ${\displaystyle \mathbb {R} }$ is not compactly generated.[5] A Hausdorff space ${\displaystyle X}$ is compactly generated if and only if it is a quotient of some locally compact space (specifically, it is a quotient of all disjoint union of all compact subspace of ${\displaystyle X}$).[5]

• 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 ${\displaystyle X}$ can be determined solely by looking at the compact (Hausdorff) subsets of ${\displaystyle X.}$ Specifically, a function ${\displaystyle f:X\to Y}$ on a (Hausdorff-)compactly generated space ${\displaystyle X}$ is continuous if and only if the same is true of its restriction ${\displaystyle f{\big \vert }_{K}:K\to Y}$ to each compact (Hausdorff) subset ${\displaystyle K\subseteq X.}$[1]
• If ${\displaystyle X}$ is (Hausdorff-)compactly generated and ${\displaystyle Y}$ is locally compact (Hausdorff), then the product ${\displaystyle X\times Y}$ is (Hausdorff-)compactly generated.
• If ${\displaystyle X}$ and ${\displaystyle Y}$ are two (Hausdorff-)compactly generated spaces, then ${\displaystyle X\times Y}$ may not be (Hausdorff-)compactly generated. Therefore, when working in categories of (Hausdorff-)compactly generated spaces it is necessary to define the product as ${\displaystyle (X\times Y)_{\operatorname {c} },}$ the k-ification of the product topology (see below).

If ${\displaystyle X}$ is compactly generated and the space ${\displaystyle C(X,Y)}$ of continuous functions has the compact-open topology then the path components in ${\displaystyle C(X,Y)}$ are precisely the homotopy equivalence classes.[5]

Given any topological space ${\displaystyle X}$ we can define a possibly finer topology on ${\displaystyle X}$ that is compactly generated, sometimes called the k-ification of the topology. Let ${\displaystyle \{K_{\alpha }\}}$ denote the family of compact subsets of ${\displaystyle X.}$ We define the new topology on ${\displaystyle X}$ by declaring a subset ${\displaystyle A}$ to be closed if and only if ${\displaystyle A\cap K_{\alpha }}$ is closed in ${\displaystyle K_{\alpha }}$ for each index ${\displaystyle \alpha .}$ Denote this new space by ${\displaystyle X_{\operatorname {c} }.}$ One can show that the compact subsets of ${\displaystyle X_{\operatorname {c} }}$ and ${\displaystyle X}$ coincide, and the induced topologies on compact subsets are the same. It follows that ${\displaystyle X_{\operatorname {c} }}$ is compactly generated. If ${\displaystyle X}$ was compactly generated to start with then ${\displaystyle X_{\operatorname {c} }=X.}$ Otherwise the topology on ${\displaystyle X_{\operatorname {c} }}$ is strictly finer than ${\displaystyle X}$ (i.e. there are more open sets).

This construction is functorial. We denote ${\displaystyle \mathbf {CGTop} }$ the full subcategory of ${\displaystyle \mathbf {Top} }$ with objects the compactly generated spaces, and ${\displaystyle \mathbf {CGHaus} }$ the full subcategory of ${\displaystyle \mathbf {CGTop} }$ with objects the Hausdorff spaces. The functor from ${\displaystyle \mathbf {Top} }$ to ${\displaystyle \mathbf {CGTop} }$ that takes ${\displaystyle X}$ to ${\displaystyle X_{\operatorname {c} }}$ is right adjoint to the inclusion functor ${\displaystyle \mathbf {CGTop} \to \mathbf {Top} .}$

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 ${\displaystyle \mathbf {CGHaus} }$ is given by ${\displaystyle \left(Y^{X}\right)_{\operatorname {c} }}$ where ${\displaystyle Y^{X}}$ is the space of continuous maps from ${\displaystyle X}$ to ${\displaystyle Y}$ with the compact-open topology.

These ideas can be generalized to the non-Hausdorff case.[3] This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.

• Compact-open topology
• Countably generated space – topological space in which the topology is determined by its countable subsetsPages displaying wikidata descriptions as a fallback
• CW complex – Type of topological space
• Finitely generated space – topological space in which the intersection of any family of open sets is openPages displaying wikidata descriptions as a fallback
• K-space (functional analysis)
• Weak Hausdorff space – concept in algebraic topologyPages displaying wikidata descriptions as a fallback

• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8.
• J. Peter May, A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics ISBN 0-226-51183-9 (See Chapter 5.)
• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
• Strickland, Neil P. (2009). "The category of CGWH spaces" (PDF).
• Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.

• 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