Simply connected space

Definition

A topological space $X$ is simply connected if it is path-connected and, for one—and hence every—basepoint $x_0\in X$ , its fundamental group is trivial:

\pi_1(X,x_0)=\{1\}.

Equivalently, every loop in $X$ can be continuously contracted to a constant loop while its basepoint remains fixed.

Basepoint and hypotheses

Path-connectedness makes fundamental groups at different basepoints isomorphic, so the definition does not depend on the chosen point. No local path-connectedness or semilocal simple-connectivity hypothesis is part of the definition; those additional conditions enter standard existence and classification theorems for covering spaces.

Examples

Convex subsets of real vector spaces, spheres $S^n$ for $n\geq2$ , and Euclidean spaces are simply connected. The circle and punctured plane are not simply connected: each has fundamental group isomorphic to $\mathbb Z$ .

References

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. Author-hosted text. Relevant: §1.1.