Category of topological spaces with continuous maps
This article describes a category (in the mathematical sense) whose objects are topological spaces, and whose morphisms are continuous maps. In other words, it gives a category structure to the collection of all topological spaces.
View other category structures on topological spaces
Definition
The category of topological spaces with continuous maps, often simply called the category of topological spaces, is defined as follows:
Aspect | Name | Definition/description |
---|---|---|
objects | topological spaces | A topological space is a set along with a collection of subsets, called open subsets, that contains the empty subset and the whole space, and is closed under taking arbitrary unions and finite intersections. |
morphisms | continuous maps between topological spaces | A continuous map of topological spaces is a set map with the property that for every open subset of , is an open subset of . |
composition law for morphisms | compose as set maps | Not needed. |
This is the default category structure on the collection of topological spaces. The category is sometimes denoted .
Constructions in this category
Construct | Name in this category | Definition/description |
---|---|---|
isomorphism | homeomorphism | A homeomorphism is a continuous bijection whose inverse is continuous. |
monomorphism | homeomorphism onto its image, which is endowed with the subspace topology | |
epimorphism | quotient map equivalent (?) |
Enhancements of this category
New category/category-like structure | Objects | Morphisms | Other aspects |
---|---|---|---|
category of based topological spaces with based maps | based topological spaces: a based topological space is a topological space with a specially marked point in it called its basepoint | based continuous maps: a continuous map of the topological spaces that sends the basepoint to the basepoint. | |
2-category of topological spaces with continuous maps and homotopies | topological spaces | continuous maps | the 2-morphisms are homotopies |
Functors from this category
Target category | Name/description of functor | Behavior on objects | Behavior on morphisms |
---|---|---|---|
category of sets with set maps | forgetful functor from category of topological spaces with continuous maps to category of sets | sends a topological space to its underlying set, i.e., forgets the topology. | sends a continuous map to the same map, now viewed as a set map between the underlying sets. |
homotopy category of topological spaces | homotopy functor from category of topological spaces to homotopy category of topological spaces | sends a topological space to itself | sends a map of topological spaces to its homotopy class. |