A wedge of \(n\) circles is a Hausdorff space \(X\) for which we have \(X=\bigcup_{k=1}^n C_k\text{,}\) where \(C_k\) is homeomorphic to \(S^1\) for all \(1\leq k\leq n\text{,}\) and there is an element \(p\in X\) such that \(C_j\cap C_k=\{p\}\) for all \(j\ne k\text{.}\)
Example2.16.2.Wedge of \(n\) cirles.
For each positive integer \(n\text{,}\) let \(C_n\subseteq \R^2\) be the circle with equation \((x-n)^2+y^2=n^2\text{.}\) For any positive integer \(n\text{,}\) the subspace \(X=\bigcup_{k=1}^n C_k\) is a wedge of \(n\) circles.
Theorem2.16.3.Wedge of \(n\) circles.
Let \(X=\bigcup_{k=1}^nC_k\) be a wedge of circles, let \(\{p\}=\bigcap_{k=1}^nC_k\text{,}\) and for each \(1\leq k\leq n\) let \(f_k\in P(C_k; p, p)\) be a loop whose path homotopy equivalence class \([f_k]\) in \(C_k\) generates \(\pi_1(C_k, p)\cong \Z\text{.}\) We have
and \(\{[f_k]\}_{k=1}^n\) is a system of free generators of \(\pi_1(X,p)\text{.}\)
Definition2.16.4.Wedge of circles.
Let \(\{X_\alpha\}_{\alpha\in I}\) be a family of subspaces of the space \(X\) satisfying \(X=\bigcup_{\alpha\in I}X_\alpha\text{.}\) The topology of \(X\) is coherent with respect to \(\{X_\alpha\}_{\alpha\in I}\) if for all subsets \(A\subseteq X\) we have \(A\) closed if and only if \(A\cap X_{\alpha}\) is closed in \(X_\alpha\) for all \(\alpha\in I\text{.}\) Equivalently, the topology is coherent with respect to the \(X_\alpha\) if for all subsets \(A\subseteq X\) we have \(A\) open if and only if \(A\cap X_{\alpha}\) is open in \(X_\alpha\) for all \(\alpha\in I\text{.}\)
A wedge of circles is a space \(X\) such that (i) \(X=\bigcup_{\alpha\in I} C_\alpha\) for some family of subspaces \(\{C_{\alpha}\}_{\alpha\in I}\) where each \(C_\alpha\) is homeomorphic to \(S^1\text{,}\) (ii) there is an element \(p\in X\) such that \(C_\alpha\cap C_{\alpha'}=\{p\}\) for all \(\alpha\ne \alpha'\text{,}\) and (iii) \(X\) is coherent with respect to \(\{C_\alpha\}_{\alpha\in I}\text{.}\)
Theorem2.16.5.Wedge of circles.
Let \(X=\bigcup_{\alpha\in I} C_\alpha\) be a wedge of circles, let \(\{p\}=\bigcap_{\alpha\in I}C_k\text{,}\) and for each \(\alpha\in I\) let \(f_\alpha\in P(C_\alpha; p, p)\) be a loop whose path homotopy equivalence class \([f_\alpha]\) in \(C_\alpha\) generates \(\pi_1(C_\alpha, p)\cong \Z\text{.}\) We have
