Prove that [0,1] and (0,1) are not homeomorphic by showing that any continuous map f: [0,1] --> (0,1) is not onto. Similarly, prove that [0,1), [0,1], (0,1) are not homeomorphic to each other.
I have no idea how to do it.
Can anyone tell me?
Think about the neighbourhoods of 0 and 1 in [0,1]. Remember, if f is continuous then every open preimage in (0,1) must have an open image in [0,1] (i.e. which open set in [0,1] contains the points 0 or 1? And, how does this show not onto?)
Are you asking for recommendations on a textbook from which to self-study topology?
If so, I recommend beginning by studying basic Metric Spaces. Consider Elementary Theory of Metric Spaces by Robert Reisel. That is written for use in a Moore type course.
If on the other hand, if you must study traditional topology, there are many good older textbooks reprinted by Dover.
A newer text by Fred. H. Croom is one I particularly like.