Based on your hint, suppose we have a homeomorphism between [0,1] and [0,1) such that

.

If h is homeomorphism, then the restriction of h removing 1 from the domain of h and h(1) from the codomain should be homeomorphism such that

.

We see that is not a homeomorphism because the domain of is connected but the codomain of is not connected. Contradiction.

Thus, h is not a homeomorphism either.

The remaining cases are similar to the above one.