A topological space \(X\) is simply connected if \(X\) is path connected and the fundamental group \(\pi_1(X, x_0)\) is trivial for every \(x_0\in X\).
For a path connected space \(X\), the following are equivalent:
\(X\) is simply connected;
every continuous map \(\gamma\colon S^1\to X\) is homotopic to a constant map;
every continuous map \(\gamma\colon S^1\to X\) extends to a continuous map \(\hat\gamma\colon D^2\to X\), that is, \(\hat\gamma|_{S^1} = \gamma\);
any two paths \(p_1, p_2\colon[0,1]\to X\) with \(p_1(0) = p_2(0)\) and \(p_1(1) = p_2(1)\) are homotopic rel endpoints.
See [@Hat02].