TITLE: Are $(0,1) \times [0,1)$ and $[0,1) \times [0,1)$ homeomorphic?
QUESTION [9 upvotes]: I do not know how to approach. I tried removing points and see if the remaining spaces are connected or not, but i couldn't conclude anything without doubt.

REPLY [2 votes]: Consider the open unit disk together with an arc of finite length from its bounding unit circle.
I think it's clear that any two such sets are homeomorphic,  independent of the length of the arc.
For the sets in question, which are squares, project (stretch) each homeomorphically from its center to the circumscribing disk. The result is two sets as above, one with a fourth and one with a half the circumference.