[[Connectedness]]
# Simple connectedness

A [[Path connectedness|path-connected]] space is **simply connected**[^einfach] or **1-simply connected** iff every [[continuous loop]] is [[Null-homotopic map|null-homotopic]], i.e. its [[fundamental group]] is trivial #m/def/homotopy 
$$
\begin{align*}
\pi_{1}(X,x_{0}) = \{ e \}
\end{align*}
$$
for any $x_{0} \in X$.

[^einfach]: German _einfach zusammenhängend_

## Examples

- Every [[Star domain]] is simply connected.
- [[Fundamental group of a sphere]]

## Properties

- Simply connectedness $\implies$ [[Local simple connectedness]] $\implies$ [[Semilocal simple connectedness]]

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