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I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity.

The question then follows. If all we know is the notion of a metric on a set, and an open set in a metric space, then what is the motivation for developing the topological definition of continuity? That is, assuming no knowledge of the results that follow such a definition, assuming no notion of epsilon-delta definition, why would anyone say:

"Hmm. Maps for which the preimage of every open set is open are good maps! I shall define them as 'continuous'. Furthermore, I shall then focus on families of subsets that contain arbitrary unions and finite intersections of their members, as well as the null and the universe, and I shall call such families topologies. I shall treat them oh so specially because these sets are just like open sets in a metric space. Oh look, it leads ro some remarkable results!"

You may think this is a natural thing to invent because you are biased with all the knowledge about the results arising from such a notion. My deep curiosity is about plain, unspoiled intuition in a virgin world, and I hope you will help me by providing your pure motivation that is independent even from the elementary epsilon-delta concept.

Think about this: they say that "neighborhood" notion helps to generalize the notion of "closeness" between elements without any reliance on a distance. Yes, this makes a lot of sense. What I don't see is why we want these neighborhoods to bear the properties of open sets???

Added: My question have been closed. However, I would like to thank Qiaochu Yuan who was remarkably open and unbiased and directed us to a prior discussion on the exact topic of my question. (See link in comment section below.)

To all others who rushed to disqualify my question, I have just found the answer. There turned out to be, in fact, a motivation for defining a topology---a motivation that does not rely on any intuition powered by the experience with Euclidean Spaces. It relies only on the mental construct of generalized "closeness" between elements described by neighborhoods (subsets). It follows remarkably that to fully describe such "closeness", it suffices to look at those subsets that bear the properties of opens sets. Precisely the topology! A notion of continuity beautifully follows. What an insight. :) But oh well, the powerful ones of this forum feel that such an approach is unworthy. :)

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    $\begingroup$ But we do know the notion of epsilon-delta continuity. $\endgroup$
    – Robin Chapman
    Commented Jul 1, 2010 at 20:19
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    $\begingroup$ This is a strange question. You want us to motivate the most general notion of continuity without appealing to our intuition for spaces like R^n; however, that is precisely the correct motivation! Maybe some people will have other opinions, but I'm pretty sure that most mathematicians use specific and concrete things (like R^n) to motivate definitions and results concerning abstract things (like topological spaces). Why would you want to do otherwise? $\endgroup$
    – Andy Putman
    Commented Jul 1, 2010 at 20:21
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    $\begingroup$ But it IS the epsilon-delta. It's a theorem in $\epsilon-\delta$ theory of continuity that the inverse image of an open set is open. Then, at some point, you start to encounter geometric objects without obvious metrics...so you need a notion of continuity that doesn't require the metric. Thus, the theorem becomes the definition. $\endgroup$
    – Charles Siegel
    Commented Jul 1, 2010 at 20:26
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    $\begingroup$ Your question is strange because it seems to assume that concepts jump out of thing air, independently of examples. Quite the contrary: the notions of continuity is an abstraction of the observed fact that some of the functions people had interest were continuous. Continuous functions very much preceded the definition of continuity! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jul 1, 2010 at 20:30
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    $\begingroup$ I think it would be a nice policy to start a count down before closing a question. This way a contributor would avoid loosing his/her time in writing a useless answer. $\endgroup$
    – Pietro Majer
    Commented Jul 1, 2010 at 21:14

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If you prefer, the "neighbourhoods" definition of continuity may be more intuitive: f is continuous if there are neighbourhoods of x whose images are contained in arbitrarily small neighbourhoods of f(x). Such functions are visibly "nice" in a metric space, since it amounts to saying that points sufficiently near x get mapped to points near f(x). Furthermore, neighbourhoods can be characterized without recourse to the metric because they clearly satisfy the unions / intersections definition for open sets, so that we can define continuity in terms of neighbourhoods without needing a metric.

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