Category of topological spaces: Difference between revisions
Created page with "The '''category of topological spaces''', denoted <math>\mathsf{Top}</math> or <math>\mathbf{Top}</math>, is the category whose objects are topological spaces and whose morphisms are continuous functions. == Definition == The category <math>\mathsf{Top}</math> consists of: * <math>\operatorname{ob}(\mathsf{Top})</math> consists of all topological spaces, * <math>\operatorname{mor}(\mathsf{Top})</math> consi..." |
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The category <math>\mathsf{Top}</math> consists of: | The category <math>\mathsf{Top}</math> consists of: | ||
* <math>\operatorname{ob}(\mathsf{Top})</math> | * <math>\operatorname{ob}(\mathsf{Top})</math> is the class of all topological spaces; | ||
* <math>\operatorname{mor}(\mathsf{Top})</math> | * <math>\operatorname{mor}(\mathsf{Top})</math> is the class of all continuous functions between topological spaces; | ||
* The composition operation <math>\circ</math> is given by the composition of ordinary functions. | |||
== Subcategories == | == Subcategories == | ||
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* <math>\mathsf{Top}_{*}</math>, [[category of pointed topological spaces]]; | * <math>\mathsf{Top}_{*}</math>, [[category of pointed topological spaces]]; | ||
* <math>\mathsf{CW}</math>, [[category of CW complexes]] or cell complexes. | * <math>\mathsf{CW}</math>, [[category of CW complexes]] or cell complexes. | ||
== See also == | |||
* [[Category theory]] | |||
* [[Morphism]] | |||
* [[Category of sets]] | |||
* [[Category of groups]] | |||
Revision as of 11:26, 6 April 2026
The category of topological spaces, denoted 
or 
, is the category whose objects are topological spaces and whose morphisms are continuous functions.
Definition
The category consists of:
is the class of all topological spaces;
is the class of all continuous functions between topological spaces;- The composition operation
is given by the composition of ordinary functions.
Subcategories
Several important subcategories of arise by restricting objects:
, 
, 
, category of pointed topological spaces;
, category of CW complexes or cell complexes.