Visualizing topological spaces

[Cincinnatus]

"visualizing" topological spaces

I am taking my first topology course right now.
My professor spends most of the time in class proving theorems that all sound like "if a space has property X then it must have property Y."

Now this is fine, but my trouble comes in finding an example of a space that actually has the various topological properties we talk about. compactness, connectedness, hausdorff etc.

For example, we often have homework problems that ask for examples of spaces that have various properties. (give an example of a topological space where points are closed that is not hausdorff; and give an example of a compact topological space which is not hausdorff are examples of these kinds of questions).

My question for you is, how do you go about answering that kind of question. What sort of thought process do you go through? It seems (to me anyway) to be a pretty different sort of question than one usually gets in a math class...

 

[homology]

Well in many math classes you're asked to find counterexamples to weakened theorems. So in a Hausdorff spaces points are closed, but there are spaces where points are closed that are not Hausdorff (R with the finite complement topology for example). I suppose there are various ways to work on problems

I always try to draw something, even if it is only a diagram of the functions and spaces present.

Read through the proof of the relevant theorem. So in this example read through the proof that in Hausdorff spaces, finite point sets are closed. Try thinking of a property like Hausdorff differently, what might it say about the topology in terms of fineness or coarseness? If R is Hausdorff, maybe a coaser topology won't be, but maybe finite point sets will still be closed.

You might also try picking up a copy of "counterexamples in topology" from the library. And maybe listing different topological spaces in the back of your notebook that are discussed in your text, your homework or in lecture.

Good luck,

Kevin

 

[Tzar]

I'm not sure if this will help but there are many obvious example of topological spaces. All metric spaces, inner product spaces, normed vector spaces, etc. are topological spaces whith the obvious topologies.

 

[Tide]

I think you should also be asking the prof for examples of such spaces as he/she is going through the proofs! :)

 

[matt grime]

Start with a list of famous topologies and see what happens.

Finite, cofinite, discrete, metric, (zariski if you're feeling brave), and don't forget you can write down a small finite set and try working out topologies on that.

both example criteria you state occur for the co-finite topology on R (zariski topology: closed sets={roots of polynomials} It is compact and points are closed and not hausdorff by a very long way.

 

[Cincinnatus]

Thanks for what everyone has said already,

So, what are some examples of spaces that are hausdorff but not regular?
What about regular but not normal?