Proving $Y\setminus\{(-1,0)\}$ using the standar topology on $\mathbb{R}^2$

Question

Consider $Y$ to be the subspace of $\mathbb{R}^2$ given by $Y=\{(x,y):x^2+y^2=1\}\cup\{(x,y):(x-2)^2+y^2=1\} $.

I want to prove that $Y\setminus\{(-1,0)\}$ is connected.

I thought of using discs in $\mathbb{R}^2$,but I am not allowed to use the concept of distance.

I try to go by contradiction by admitting that there is a clopen set in the subspace $Y\setminus{(-1,0)}$ but that lead me nowhere.

Question:

How should I prove $Y\setminus{(-1,0)}$ is connected?

Thanks in advance!

Answer 1

$\{(x,y) :x^{2}+y^{2}=1\}\setminus (-1,0)$ is connected and so is $\{(x,y) :(x-2)^{2}+y^{2}=1\}\setminus(-1,0)$. Since the point $(1,0)$ is common to these, their union is connected.