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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...

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# Visualizing topological spaces

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