Simple connectedness implies semilocal simple connectedness

If \(X\) is simply connected, then \(X\) is path connected and \[ \pi_1(X,x)=0 \] for every basepoint \(x\in X\). Let \(x\in X\). Taking \(U=X\) itself (or any path-connected neighborhood of \(x\)), the inclusion-induced homomorphism \[ \pi_1(U,x)\longrightarrow\pi_1(X,x) \] has trivial target and hence is trivial. This is precisely the semilocal simple-connectedness condition. Therefore every simply connected space is semilocally simply connected.