Let \(0\leq n\leq\infty\text{.}\) A map \(e\colon Y\to Z\) is an \(n\)-equivalence if \(\pi_0(Y)\to\pi_0(Z)\) is surjective for \(n=0\) and bijective for \(n>0\text{,}\) and for every \(y\in Y\) the map \(e_*\colon\pi_q(Y,y)\to\pi_q(Z,e(y))\) is an isomorphism for \(1\leq q<n\) and a surjection for \(q=n\geq1\text{.}\) An \(\infty\)-equivalence is called a weak homotopy equivalence.
A nonempty space \(X\) is \(n\)-connected if \(\pi_q(X, x) = 0\) for all \(0 \leq q \leq n\) and all \(x \in X\text{.}\) A pair \((X, A)\) with \(A \neq \varnothing\) is \(n\)-connected if \(\pi_0(A) \to \pi_0(X)\) is surjective and \(\pi_q(X, A, a) = 0\) for all \(1 \leq q \leq n\) and all \(a \in A\text{.}\)
A space is \(0\)-connected if and only if it is path-connected, and \(1\)-connected if and only if it is simply connected. For a pair, the long exact sequence
\begin{equation*}
\pi_q(A, a) \to \pi_q(X, a) \to \pi_q(X, A, a) \to \pi_{q-1}(A, a) \to \pi_{q-1}(X, a)
\end{equation*}
shows that \((X, A)\) is \(n\)-connected if and only if the inclusion \(A \hookrightarrow X\) is an \(n\)-equivalence.
