Thread: Topology

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?)

Since I watch the book myself, I do not quite understand the topology.
Can you recommend books to me?

Originally Posted by DEviL

Since I watch the book myself, I do not quite understand the topology.
Can you recommend books to me?

Truly I have no idea what you mean by that post.
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.

Originally Posted by DEviL

Since I watch the book myself, I do not quite understand the topology.
Can you recommend books to me?

I recommend not just books, but lots and lots of practice with rudimentary real analysis and set theory. Sometimes you just need lots of examples to fill your "bag of tricks". You can find all of these online without the purchase of textbooks. Search for worked examples.

Thank you very much!
I bought a book which only has the definition of some terms and exercise so I want to have some illustrations about it.