# Definition of topology

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

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.