TITLE: Let $X=\mathbb{R^2}-\{0\}$ and $Y=S^1 \cup\{[0,1]\times\{0\}\}$. Prove or Disprove that $X$ is homeomorphic to $Y$.
QUESTION [5 upvotes]: Let $X=\mathbb{R^2}-\{0\}$ and $Y=S^1 \cup\{[0,1]\times\{0\}\}$.
Prove or Disprove that $X$ is homeomorphic to $Y$. 
My thoughts:
Since X is the punctured plane, in terms of retraction, I believe that $S^1$ is a deformation retact of $X$, Similarly, the space $Y$ is the unit circle with an extra "strip" along the $x- axis$ same as the radius. And so  $S^1$ is a deformation retract of $Y$ as well. 
As a result, the fundamental groups of both $X$ and $Y$ are  isomorphic to the fundamental group of the unit circle $\pi_1(X,x_0) \approx \mathbb{Z} \approx \pi_1(Y,y_0)$ .
Does this isomorphism guarantee a homeomorphism between the spaces $X$ and $Y$?
if not, Is there a way to appraoch it without involving fundamental groups?

REPLY [2 votes]: $HINT:$ Deleting the point $(1,0)$ from $Y$ results it disconnected. Can you find such point in $X$?