[[Naïve set theory MOC]]
# Constant map

A **constant map** $cT : X \to Y$ is a map whose range $cTX$ consists of a single element $c*$. #m/def/general 
A map is constant iff it can be factored via the [[single point space]],
hence it always has the form $cT$ where $T : X \to *$ is the unique morphism.

## Properties

### Topology

- A [[Homotopy of maps|homotopy]] between constant maps is a [[continuous path]]. Therefore, a space is [[Path connectedness|path-connected]] iff all constant maps form a single [[Homotopy of maps|homotopy class]].
- A map is null-homotopic iff it is homotopy to a constant map

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#state/tidy | #lang/en | #SemBr