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I am learning the topology from the book by Munkres. Munkres starts up the topic by describing the way topology was defined. It says that whensoever we define anything in mathematics we define it in such a way that it covers some interesting aspects of mathematics that can be studied under that object being defined and at the same time it should be restricted from being over general.

Can anyone shed some light on the way the definition of topology was formulated along the lines aforementioned? May I know the difference between point set topology and general topology?

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Point set topology and general topology are just two names for the same thing; there is no real difference. – Brian M. Scott Dec 21 '12 at 10:03
Yes. Point-set topology is general topology. To be distinguished from algebraic topology, differential topology, etc. – Hui Yu Dec 21 '12 at 15:19
@HuiYu Thank you – danny gotze Dec 22 '12 at 9:48
@BrianM.Scott Sir, will it be right to say that we investigate the same things in topology as in analysis but in a quasi quantitative way of open sets from which we derive general properties of metric spaces? – danny gotze Dec 22 '12 at 9:49
Not really: metric spaces are just a small part of topology. It would be better to say that general topology deals with concepts that have their roots in metric spaces but that generalize those roots enormously, mostly in directions that move away $-$ sometimes very far away $-$ from the quantitative aspects of metric spaces. – Brian M. Scott Dec 22 '12 at 9:55

A Topology comes from the set of neighborhoods of points. When you're doing calculus (infinitesimal calculus - derivatives and limits of functions), say, looking at a limit of a function at a point. You're looking at what happens to value of the function when you're delving closer and closer to a given point - that is, you're looking at the relation between the set of values of the function and the set of neighborhoods of the point. All of calculus is built on such considerations. By studying topology, you can redefine problems in calculus in a way that makes them much more simple, and topology then allows you to try and do similar things on spaces which are less convenient than the Euclidean Spaces.

An example - think about the definition of an equi-continuous function. The $\delta-\epsilon$ definition is annoying. When you define it in terms of small open sets, rather then epsilons and deltas, the outcome is a beautiful and revealing definition which actually tells you something intuitive about the function.

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Thanks for your answer. I hope your answer is rich with insights whose cream I can capture when I will go through the mentioned function in terms of topology. – danny gotze Dec 21 '12 at 10:28
While i would agree with the answer, i know numerous persons who are very happy when dealing with epsilons and deltas... ;) – mkl Dec 21 '12 at 11:18
But epsilons and deltas only work when you have a metric. They won't take you far. – mousomer Dec 21 '12 at 12:23

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