[[Simple connectedness]]
# Semilocal simple connectedness

A topological space $X$ is called **semilocally simply connected** iff every $x \in X$ has an open neighbourhood $U \sube X$ so that every [[continuous loop]] in $U$ is [[Null-homotopic map|null-homotopic]] in $X$,
#m/def/topology 
i.e. if $\iota: (U,x) \hookrightarrow (X,x)$ is the natural inclusion then $\pi_{1}\iota : \pi_{1}(U,x) \to \pi_{1}(X,x)$ is the trivial homomorphism.

## Properties

- [[A locally path-connected, connected space is semilocally simply connected iff it has a universal covering]]

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