[[Topology MOC]]
# Category of topological spaces

The **category of topological spaces** $\Top$ is the [[category]] where each object is a [[topological space]] and each morphism is a [[Continuity|continuous map]]. #m/def/topology 

> [!check]- Well defined category
> - Composition of continuous maps yields a continuous map
> - The identity is always continuous
> - Function composition is already associative

## Properties

### Universal constructions

- The [[Initial and terminal objects|initial object]] of $\Top$ is $\0$ with the unique topology
- The [[Initial and terminal objects|terminal object]] of $\Top$ is $\{ * \}$ with the unique topology

### Morphisms

- [[Regular monomorphisms in the category of topological spaces]]

## Related categories

- [[Naïve homotopy category]] is the [[Quotient category]] of $\hTop$ with [[Homotopy of maps]].

#
---
#state/develop | #lang/en | #SemBr