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 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 **compactly generated Hausdorff space** is a compactly generated space that is also Hausdorff. Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff or weakly 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 *n*Lab on 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.^{ [2] } 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 compactly generated (WHCG) spaces.

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 , 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 is a non-principal ultrafilter on an infinite set , the induced topology has the property that every compact set is finite, and is not compactly generated.

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.

Given any topological space *X* we can define a (possibly) finer topology on *X* that is compactly generated. 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 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 right adjoint to the inclusion functor **CGTop** → **Top**.

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

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In topology and related areas of mathematics, a **product space** is the Cartesian product of a family of topological spaces equipped with a natural topology called the **product topology**. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In topology and related branches of mathematics, **Tychonoff spaces** and **completely regular spaces** are kinds of topological spaces. These conditions are examples of separation axioms.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis.

In topology and related branches of mathematics, a topological space is called **locally compact** if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

In mathematics, a **base** or **basis** for the topology τ of a topological space (*X*, τ) is a family *B* of open subsets of *X* such that every open set is equal to a union of some sub-family of *B*. For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In mathematics, a **Baire space** is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.

In mathematics, an **order topology** is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

In the mathematical discipline of general topology, a **Polish space** is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

In mathematics, the **category of topological spaces**, often denoted **Top**, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of **Top** and of properties of topological spaces using the techniques of category theory is known as **categorical topology**.

In mathematics, the **compact-open topology** is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945.

In topology and related areas of mathematics, a **subspace** of a topological space *X* is a subset *S* of *X* which is equipped with a topology induced from that of *X* called the **subspace topology**.

In functional analysis and related areas of mathematics, a **barrelled space** is a topological vector spaces (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A **barrelled set** or a **barrel** in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

In topology and related fields of mathematics, a **sequential space** is a topological space that satisfies a very weak axiom of countability.

In mathematics, a topological space *X* is said to be **limit point compact** or **weakly countably compact** if every infinite subset of *X* has a limit point in *X*. This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.

In topology, a **coherent topology** is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a *topological union* of those subspaces. It is also sometimes called the **weak topology** generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps.

- ↑ Hatcher, Allen (2001).
*Algebraic Topology*(PDF).*(See the Appendix)* - 1 2 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.

- Compactly generated spaces - contains an excellent catalog of properties and constructions with compactly generated spaces
- Compactly generated topological space in
*nLab* - Convenient category of topological spaces in
*nLab*

- 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).

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