I know how to show $[0,1] \times [0,1]$ is not homeomorphic to $(0,1) \times (0,1)$ by a compactness argument. Is there such an argument that shows $[0,1) \times (0,1)$ is not homeomorphic to $(0,1) \times (0,1)$? If not, what is the best way to show that they’re not homeomorphic?
In both spaces every point has a local basis consisting of simply connected open environments. In the latter space, removing the point from those simply connected environments always leads to a set that is no longer simply connected. In the former space there exist points (which we could call the “boundary” if it were not such a charged term in a topology course) where removing the point from the simply connected open environment leaves a simply connected set.
Come to think of it: in the former space there exist points whose complements are simply connected. Not so in the latter.
Consider the $1$ point compactification of each space. For $[0,1)\times (0,1)$ it’s the closed disk, for $(0,1) \times (0,1)$ it’s the sphere.