On 3-manifolds that are not simply connected

Let X be a connected 3-manifold.Then X is simply connected if and only if each simple closed curve in X lies in a simply connected neighborhood.On the other hand, each point of X trivially lies in a simply connected neighborhood with or without X itself simply connected.However, we will show that arcs are big enough to reflect the global property of X as to simple connectedness in the sense that X is simply connected if and only if each arc in X lies in a simply connected neighborhood.