Yes/ No Is $X$ is homeomorphics to $Y$?

$X={(x,y) in mathbb{R^2}: x^2+y^2=1 }$ and $Y={(x,y) in mathbb{R^2}: x^2+y^2=1 } cup {(x,y) in mathbb{R^2}: (x-2)^2+y^2=1 } $ be the subspaces of $mathbb{R}^2$

Now my question is that

Is $X$ is homeomorphics to $Y$ ?

My attempt : Yes

Both $X$and $Y$ are connected and compact

Now by using the theorem connected subspaces of connected sets is connected

So $X$ is homeomorphics to $Y$

Is its true ?