Semilocally simply connected space

Definition
A topological space is said to be semilocally simply connected or semilocally 1-connected if it satisfies the following equivalent conditions:


 * 1) For any $$x \in X$$ there exists an open subset $$U \ni x$$ such that the homomorphism of fundamental groups induced by the inclusion of $$U$$ in $$X$$ is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: $$\! \pi_1(U,x) \to \pi_1(X,x)$$. Note that if $$U$$ and/or $$X$$ are not connected, we interpret the fundamental groups as referring to the fundamental groups of the path components of $$x$$ in the respective subsets.
 * 2) For any $$x \in X$$ and any open subset $$V$$ of $$X$$ containing $$x$$, there exists an open subset $$U \ni x$$ such that $$U \subseteq V$$ and the homomorphism of fundamental groups induced by the inclusion of $$U$$ in $$X$$ is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: $$\! \pi_1(U,x) \to \pi_1(X,x)$$. In other words, every loop about $$x$$ contained in $$U$$, is nullhomotopic in $$X$$. Note that if $$U$$ and/or $$X$$ are not connected, we interpret the fundamental groups as referring to the fundamental groups of the path components of $$x$$ in $$U$$ and $$X$$ respectively.

Note that the term is typically used for spaces that are locally path-connected spaces. In this case, we can assume that the open subset $$U$$ is path-connected.

Stronger properties

 * Locally simply connected space
 * Simply connected space

Textbook references

 * , Page 494 (formal definition)
 * , Page 297 (formal definition): Introduced as semilocally 1-connected
 * , Page 63 (formal definition)
 * , Page 78 (forma definition): Introduced as semilocally 1-connected