I need some help with the the following problem. I am not aware what technique should I use to solve the problem:
In $\mathbb{R}^2$, we denote by $S_{x,y}$ a unit circle with center at $(x,y) \in \mathbb{R}^2 $. Let $X = S_{-1,0}\cup S_{1,0} \cup S_{0,\sqrt{3}}; \quad Y = S_{-2,0}\cup S_{0,0} \cup S_{2,0}; \text{ and } Z $ the bouquet of 3 circles.
(Q-1)Which of them are homeomorphic to each other?
(Q-2)which of them are homtotopy equivalent to each other?
It does feel like they are all homtopy equivalent to each other but none of them is homemorphic to each other, but I don't know what tool to use to prove them.
Q-3) Can we use Fundamental group to solve this problem? I have a feeling that we can use Van Kampen theorem to solve the problem but I haven't read Van Kampen's theorem yet.