---
tag: K1DX0S4797
kind: example
name: Category of topological spaces
msc: [18F60, 54B30]
instance_of: [A2DE66937D]
uses: [0CQHFPGJ45, 0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Category_of_topological_spaces
same_as:
  wikidata: Q2000555
  wikipedia: Category of topological spaces
---

## Example

The *category of topological spaces* $\mathbf{Top}$ has the [[topological spaces|0CQHFPGJ45]] as objects and the [[continuous maps|0VGFHTM75Y]] as morphisms.
Composition is the composition of maps, and the identity of $X$ is $\id_X$.
This is a [[category|A2DE66937D]].
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|FWGPKXCPB7]] 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|BFBNQ36QS8]]; for this reason algebraic topology often works in the full subcategory of compactly generated spaces.

## Explanations

### L1 human

The category of topological spaces gathers all spaces and all continuous maps between them.
Its isomorphisms are the homeomorphisms, the maps that topology regards as "the same shape".
For example, the map $t \mapsto (\cos 2\pi t, \sin 2\pi t)$ from $[0, 1]$ to the circle is a morphism, though not an isomorphism, since it glues the two ends of the interval.