A space
X is called
semi-locally simply connected if every
point x in
X has a
neighborhood U with the property that every
loop in
U based at
x can be
contracted within
X to the constant loop at
x (i.e., every loop in
U starting and ending at
x is
nullhomotopic in
X via a basepoint-preserving homotopy). Note that if
U satisfies this condition, so does any smaller neighborhood of
x, so that
x has arbitrarily small neighborhoods satisfying the condition. The neighborhood
U need not be
simply connected: though every loop in
U based at
x must be contractible within
X, the contraction is not required to take place inside of
U. For this reason, a space can be semi-locally simply connected without being
locally simply connected. Also, it is not required that every loop in
U is nullhomotopic in
X; it is only the loops in
U based at
x that must be nullhomotopic in
X. In general, a semi-locally simply connected space may have points
x with arbitrarily small neighborhoods containing loops (not going through
x) that cannot be contracted to a point, even with homotopies in
X. An equivalent formulation of the definition is that every point in x\in X has an open neighborhood U for which the
homomorphism \pi_1(U,x)\to\pi_1(X,x)
induced by the
inclusion map of U into X is trivial. Here, \pi_1(U,x) is the
fundamental group of U relative to the basepoint x; and similarly for \pi_1(X,x). Most of the main theorems about
covering spaces, including the existence of a universal cover and the Galois correspondence, require a space to be
path-connected,
locally path-connected, and semi-locally simply connected, a condition known as
unloopable (
délaçable in French). In particular, this condition is necessary for a locally path-connected space to have a simply connected covering space. ==Examples==