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It's a fantastic textbook and easy to read (and cheap, if you buy the electronic copy - the best £5 I've spent). Ideally what you'd do is calculate the equivalent subgroupoid $\Pi_1(S^1,{a,b,c})$ \Pi_1(S^1,\{a,b,c\})$ where$a,b,c$are three points in$S^1$, one in each intersection of opens. 1 I can only point to the place where this was originally done (or rather, the latest edition thereof): Topology and Groupoids by Ronnie Brown It's a fantastic textbook and easy to read (and cheap, if you buy the electronic copy - the best £5 I've spent). Ideally what you'd do is calculate the equivalent subgroupoid$\Pi_1(S^1,{a,b,c})$where$a,b,c$are three points in$S^1\$, one in each intersection of opens.