Category theory; homological algebra Special categories Example

Category of topological spaces

Builds on Topological space Continuous map

Example

The category of topological spaces $\mathbf{Top}$ has the topological spaces as objects and the continuous maps as morphisms. Composition is the composition of maps, and the identity of $X$ is $\id_X$ . This is a category. If $f \colon X \to Y$ and $g \colon Y \to Z$ are continuous and $U \subseteq Z$ is open, then $(g \circ f)^{-1}(U) = f^{-1}(g^{-1}(U))$ is open, so $g \circ f$ is continuous. The identity map is continuous, and composition of maps is associative.

Remarks

The isomorphisms of $\mathbf{Top}$ are the homeomorphisms. A continuous bijection need not be an isomorphism. The identity map from $\R$ with the discrete topology to $\R$ with the usual topology is continuous and bijective, but its inverse is not continuous. $\mathbf{Top}$ is a complete category and is also cocomplete: the product is the product topology, and the coproduct is the disjoint union topology. The forgetful functor to sets has a left adjoint (the discrete topology) and a right adjoint (the indiscrete topology). $\mathbf{Top}$ is not Cartesian closed; for this reason algebraic topology often works in the full subcategory of compactly generated spaces.

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Tag	`K1DX0S4797`
URI	`m2://K1DX0S4797`
Kind	example
Writing	W0 stub
MSC	18F60, 54B30
Instance of	Category
Uses	Topological space, Continuous map
Same as	Wikidata Q2000555 Wikipedia Category of topological spaces
Changed	2 November 2026

Category of topological spaces

Example

Remarks

Reading path

Further reading

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