Let \((X_\alpha, x_\alpha)_{\alpha\in A}\) be a family of based topological spaces. The wedge sum \(\bigvee_{\alpha\in A} X_\alpha\) is the quotient space of the disjoint union \(\coprod_{\alpha\in A} X_\alpha\) obtained by identifying all the basepoints \(x_\alpha\) to a single point, which is its basepoint. For two based spaces, \(X\vee Y\) is the quotient of \(X\sqcup Y\) identifying \(x_0\) with \(y_0\).
Let \(X\) be a CW complex with \(n\)-skeleton \(X^n\). Then \(X^n/X^{n-1}\) is homeomorphic to \(\bigvee_\alpha S^n_\alpha\), a wedge of \(n\)-spheres with one sphere for each $n$-cell of \(X\).
If each \((X_\alpha, x_\alpha)\) is a good pair, then the inclusions \(X_\alpha\hookrightarrow\bigvee_\alpha X_\alpha\) induce an isomorphism \[ \bigoplus_\alpha\tilde H_n(X_\alpha)\to\tilde H_n\qty{\bigvee_\alpha X_\alpha} \] for every \(n\).
See [@Hat02].