Semilocal simple connectedness

A topological space $𝑋$ is called semilocally simply connected iff every $𝑥 \in 𝑋$ has an open neighbourhood $𝑈 \subseteq 𝑋$ so that every continuous loop in $𝑈$ is null-homotopic in $𝑋$ , #m/def/topology i.e. if $𝜄 : (𝑈, 𝑥) ↪ (𝑋, 𝑥)$ is the natural inclusion then $𝜋 1 𝜄 : 𝜋 1 (𝑈, 𝑥) \to 𝜋 1 (𝑋, 𝑥)$ is the trivial homomorphism.