A locally simply connected space and fundamental groups of one point unions of cones

Abstract:

Let CX be the cone over a space $X$. Let a space $X$ be first countable at $x$, then the following are equivalent: (1) $X$ is locally simply connected at $x$; (2) ${\pi _1}\left ( {\left ( {X,x} \right ) \vee \left ( {X,x} \right ),x} \right )$ is naturally isomorphic to the free product ${\pi _1}\left ( {X,x} \right ) * {\pi _1}\left ( {X,x} \right )$; (3) ${\pi _1}\left ( {\left ( {CX,x} \right ) \vee \left ( {CX,x} \right ),x} \right )$ is trivial. There exists a simply connected, locally simply connected Tychonoff space $X$ with $x \in X$, such that $\left ( {X,x} \right ) \vee \left ( {X,x} \right )$ is not simply connected.