Are these subsets homeomorphic? Are the two subsets of the Euclidean Plane $[0,1]\times[0,1)$ and
$[0,1)\times[0,1)$ homeomorphic or not?
My attempt:
We need to find a bijective function $f$ from $[0,1]$ to $[0,1)$ such that $f$ and $f^{-1}$ are both continuous. If $f$ is continuous then for $U\subseteq [0,1)$, we have that $f^{-1}(U)$ is open in $[0,1]$ if $U$ is open in $[0,1)$. Using the identity map and any open subset of $[0,1)$ we see that the pre-image is open in $[0,1]$ so they are homeomorphic.
Does this make sense? 

They're homeomorphic since both of them are homeomorphic to the closed unit disc in $R^2$  with an arc removed on its boundary.
