Let $X$ be the unit circle in $\mathbb{R}^2$; that is $X=\{(x,y):x^2+y^2=1\}$ and has the subspace topology. 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\} $.
Is $Y$ homeomorphic to the space $X$?
I know I could use path-connectedness to prove $X$ and $Y$ are not homeomorphic but I am not supposed to. I thought of cardinality, but I really have no clue on how to approach this problem.
Question:
How should I prove $X$ and $Y$ are not homeomorphic?
I don't know if this is what you have in mind whan you claim that you “could use path-connectedness to prove $X$ and $Y$ are not homeomorphic”; if it is, I will delete it.
If you remove a point from $X$, what remains is connected. But if you remove $(1,0)$ from $Y$, what remains becomes disconnected.