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I'm studying topology at this moment and i'm wondering if someone know where to find a summary of something of this kind that gives a visual intepretation of the standard topological properties. For example, Hausdorff, Compactness, connectedness etc. This would give a lot more intuitive meaning to the abstract definitions that are given (without pictures). Also i'm trying to develop some basic intuition for this course, which so far does not really work for me. Also the course abstract algebra I really found easy but with this course i'm really struggling. How did you guys develop the intuition for it?

Thanks a lot!

user112167
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  • http://math.stackexchange.com/questions/40338/i-need-visual-examples-of-topological-concepts?rq=1 – Newb Dec 13 '13 at 22:42
  • I saw this post, but it does not give a visual intepretation of the properties. – user112167 Dec 13 '13 at 22:48
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    Visual interpretations of the separation axioms (e.g., Hausdorff) are straightforward and helpful, but good visual interpretations of many of the more complicated properties like compactness and connectedness are essentially non-existent: any picture is almost certain to be more misleading than helpful. Ultimately you develop intuition for them by working with them. Topologists do make diagrams to help themselves picture specific spaces, but that’s a different matter, and the diagrams are sometimes quite idiosyncratic. – Brian M. Scott Dec 13 '13 at 22:51

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It is helpful to keep in mind several examples of topological spaces ($R^n$, finite-closed, some finite topologies, the spectrum of a ring, certain subsets of $R^n$, etc.) and see how these properties manifest in them. No one picture will work for all situations, but the intuition you develop in euclidean space is very very useful.

Elle Najt
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