This is problem 7-15 in Lee's Introduction to Topological Manifolds. As a hint, he suggests using the fact that a homeomorphism exists taking any n distinct points to any another n distinct points for connected manifolds. How to use the hint given?
2026-03-27 11:49:07.1774612147
Proving three circles of radius 1 centred at $(0,0), (2,0), (4,0)$ homotopy equivalent to $^1∨^1∨^1$
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I'm not sure how to use the hint so let me propose an alternative solution.
Let $Y$ be your three circles in $\Bbb R^2$. Check that contracting an arc between $(1,0)$ and $(3,0)$ is an homotopy equivalence, for example using Van Kampen's theorem. Now notice that the resulting space is $S^1 \vee S^1 \vee S^1$.