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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.2012-12-21
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    Yes. Point-set topology is general topology. To be distinguished from algebraic topology, differential topology, etc.2012-12-21
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    @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?2012-12-22
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    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.2012-12-22
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    @BrianM.Scott At the same ,when we are at it, I would like to make sure whether I correctly understand open sets and closed sets. As the common definition goes open sets are the sets for each of whose element there exist a neighborhood whose points belong to the same set and closed sets are sets whose complement are open sets,i.e. for any point outside the set there exists a neighborhood all of whose points are outside the set. Then I came through a proposal that there can be sets which can be simultaneously closed as well as open.2012-12-22
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    @BrianM.Scott And then I came across the proposal that the definition of open set depends on your choice and topology ? I could not understand what this means. Hope you can explain and elaborate this point.2012-12-22
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    If $X$ is a set, a family $\tau$ of subsets of $X$ is a *topology* on $X$ if $\varnothing,X\in\tau$, $U\cap V\in\tau$ whenever $U,V\in\tau$ ($\tau$ is closed under finite intersections), and $\bigcup\mathscr{U}\in\tau$ whenever $\mathscr{U}\subseteq\tau$ ($\tau$ is closed under arbitrary unions). By definition the open sets in the space $\langle X,\tau\rangle$ are the members of $\tau$. You’re right about the closed sets: by definition they’re the complements of open sets. And yes, it’s possible (in some spaces) for a set to be both open and closed; if $\tau=\wp(X)$, every set is both.2012-12-22
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    @BrianM.Scott Can I beg a bit more of simplicity ? I mean ,given the definition of topology, as a family of subsets which is such that the intersection and union of its elements are included in it,with the whole set and null set being compulsory inclusions, how can the definition of an open set depend on this family of subsets? Let us consider a circle in a plane. Now the interior region of the circle , is an open set as for all the points inside we can get a neighborhood containing the points of it. Whatsoever collection of subsets of the set of points inside the circle you take, its open.2012-12-23
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    @BrianM.Scott (contd.. from the previous comments) irrespective of your choice of subsets.2012-12-23
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    That family of subsets **is** the family of open sets in that space. In most spaces, however, it’s easier to describe the open sets by describing a [*base*](http://en.wikipedia.org/wiki/Base_%28topology%29) for the topology and then defining $\tau=\left\{\bigcup\mathscr{U}:\mathscr{U}\subseteq\mathscr{B}\right\}$. In $\Bbb R^2$ the collection of all sets of the form $B(x,r)=\{y\in\Bbb R^2:d(x,y) is a base for the usual topology (where $d$ is the usual distance between points): the open sets in $\Bbb R^2$ are the sets that are unions of these open balls.2012-12-23

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