Are these spaces path connected??

Question

I have some question on studying path-connected

$$S_1:=\{(x,y) \in \Bbb R^2 | (x-2)^2+y^2 \le 1\} \cup \{(x,y) \in \Bbb R^2 | (x+2)^2+y^2 \le 1\} \cup \{(x,y) \in \Bbb R^2 | -1 \le x \le 1 ,y=0\}$$

$$S_2:=\{(x,y) \in \Bbb R^2 | (x-1)^2+y^2 \le 1\} \cup \{(x,y) \in \Bbb R^2 | (x+1)^2+y^2 \le 1\} $$

Those two space are path conneced? I think we can visually draw a path on these space without rigorous proof .

Answer 1

The following two pictures should help:

$S_1$

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$S_2$

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