Compactly generated space
Encyclopedia
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:
Equivalently, one can replace closed with open
in this definition. If X is coherent with any cover of compact subsets in the above sense then it is, in fact, coherent with all compact subsets.
A compactly generated Hausdorff space is a compactly generated space which is also Hausdorff
. Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff.
. The category of topological spaces
, Top, is defective in the sense that it fails to be a cartesian closed category
. There have been various attempts to remedy this situation, one of which is to restrict oneself to the full subcategory of compactly generated Hausdorff spaces. This category is, in fact, cartesian closed. A definition of the exponential object is given below.
These ideas can be generalised to the non-Hausdorff case, see section 5.9 in the book `Topology and groupoids' listed below. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.
with objects the compactly generated spaces, and CGHaus the full subcategory of CGTop with objects the Hausdorff separated spaces.
Given any topological space X we can define a (possibly) finer topology on X which is compactly generated. Let {Kα} 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 ∩ Kα is closed in Kα for each α. Denote this new space by Xc. One can show that the compact subsets of Xc and X coincide and the induced topologies are the same. It follows that Xc is compactly generated. If X was compactly generated to start with then Xc = X otherwise the topology on Xc is strictly finer than X (i.e. there are more open sets).
This construction is functorial. The functor from Top
to CGTop which takes X to Xc is right adjoint
to the inclusion functor CGTop → Top
.
The continuity of a map defined on 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 the CGHaus is given by (YX)c where YX 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, see section 5.9 in the book `Topology and groupoids' listed below. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a compactly generated space (or k-space) is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
whose topology is coherent
Coherent topology
In topology, a coherent topology is one 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.-Definition:...
with the family of all compact subspaces
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. Specifically, a topological space X is compactly generated if it satisfies the following condition:
- A subspace A is closedClosed setIn geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
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
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
in this definition. If X is coherent with any cover of compact subsets in the above sense then it is, in fact, coherent with all compact subsets.
A compactly generated Hausdorff space is a compactly generated space which is also Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
. Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff.
Motivation
One of the primary motivations for studying compactly generated spaces comes from category theoryCategory theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. The category of topological spaces
Category of topological spaces
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...
, Top, is defective in the sense that it fails to be a cartesian closed category
Cartesian closed category
In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in...
. There have been various attempts to remedy this situation, one of which is to restrict oneself to the full subcategory of compactly generated Hausdorff spaces. This category is, in fact, cartesian closed. A definition of the exponential object is given below.
These ideas can be generalised to the non-Hausdorff case, see section 5.9 in the book `Topology and groupoids' listed below. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.
Examples
Most topological spaces commonly studied in mathematics are compactly generated.- Every compact space is compactly generated.
- Every locally compact spaceLocally compact spaceIn 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.-Formal definition:...
is compactly generated. - Every first-countable spaceFirst-countable spaceIn topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis...
is compactly generated. - Topological manifoldTopological manifoldIn mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
s are locally compact Hausdorff and therefore compactly generated Hausdorff. - Metric spaceMetric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s are first-countable and therefore compactly generated Hausdorff. - Every CW complexCW complexIn topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
is compactly generated Hausdorff.
Properties
We denote CGTop the full subcategory of TopCategory of topological spaces
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...
with objects the compactly generated spaces, and CGHaus the full subcategory of CGTop with objects the Hausdorff separated spaces.
Given any topological space X we can define a (possibly) finer topology on X which is compactly generated. Let {Kα} 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
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
A ∩ Kα is closed in Kα for each α. Denote this new space by Xc. One can show that the compact subsets of Xc and X coincide and the induced topologies are the same. It follows that Xc is compactly generated. If X was compactly generated to start with then Xc = X otherwise the topology on Xc is strictly finer than X (i.e. there are more open sets).
This construction is functorial. The functor from Top
Category of topological spaces
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...
to CGTop which takes X to Xc is right adjoint
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
to the inclusion functor CGTop → Top
Category of topological spaces
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...
.
The continuity of a map defined on 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
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is continuous when restricted to each compact subset K ⊆ X.
If X and Y are two compactly generated spaces the product
Product topology
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...
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
Exponential object
In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories...
in the CGHaus is given by (YX)c where YX is the space of continuous maps from X to Y with the compact-open topology
Compact-open 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...
.
These ideas can be generalised to the non-Hausdorff case, see section 5.9 in the book `Topology and groupoids' listed below. This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.
See also
- compact-open topologyCompact-open topologyIn 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...
- CW complexCW complexIn topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
- finitely generated space
- countably generated spaceCountably generated spaceIn mathematics, a topological space X is called countably generated if the topology of X is determined by the countable sets in a similar way as the topology of a sequential space by the convergent sequences....
- Weak Hausdorff spaceWeak Hausdorff spaceIn mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. In particular, every Hausdorff space is weak Hausdorff....