Let \(n\geq 0\). A nonempty topological space \(X\) is \(n\)-connected if \(X\) is path-connected and \(\pi_i(X, x_0) = 0\) for all \(1\leq i\leq n\) and all \(x_0\in X\). A pair \((X, A)\) with \(A\subseteq X\) is \(n\)-connected if every path component of \(X\) meets \(A\) and the relative homotopy groups satisfy \(\pi_i(X, A, x_0) = 0\) for all \(1\leq i\leq n\) and all \(x_0\in A\).
A space is \(0\)-connected if and only if it is path-connected, and \(1\)-connected if and only if it is simply connected.
[@Hat02].