Adam Frank: Also, I'll take the opportunity here to say what I think Topology is all about in outline--and if you want to contribute something more insightful, please do.

This is probably ahistorical, but here is how I imagine Topology became a subject of study: First we studied the derivative, and in doing so looked at a limit. The limit at a point involves the notion of "closeness", since it is basically "The function can be made arbitrarily close to the limit value, by making the argument appropriately close to the point". In higher dimensions this was no challenge, just using higher dimensional versions of the Euclidean distance metric.

But the we started looking at the closeness of whole functions, like with point-wise and uniform convergence. At that point (or maybe later, I don't know) we decided that we wanted a formal study of the idea of closeness. And that is nearly what a topology is--it is a way of talking about which points are close to other points. This is particularly true in metrizable spaces, and even still kind of feels true in Hausdorff spaces, and feels significantly less true in non-Hausdorff spaces. But in non-Hausdorff spaces, I'm not sure it's even possible to have an intuitive feel for what these things "really are". It seems like, with those, you just have to accept them as formal definitions that have surprisingly useful applications.

Another perspective on what topology is, is to consider spaces where different subsets of points are considered "the same shape" if one shape can be continuously deformed into the other--so that basically you have the notion of a shape while removing the notion of a distance metric. I don't feel that I 100% understand how that connects to the other way of seeing topology as being "about closeness", but maybe by the end of the book I will.

Zohreh Jafari: Thanks for the comment. Seeing Topology through "closeness" is an analytic view on the subject.

Topology is all about "studying shapes". Its formal definition usually goes like:

*Topology is the mathematical study of the properties that are preserved through deformations, twisting, and stretching of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid.*

It seems, from the beginning, manipulation of numbers(quantity) has been easier for mathematicians and (maybe I don't know) engineers. Beside four basic operations that we use on daily basis, important concepts of equivalence and dimension naturally come to us. Also, there are many ways that we can combine numbers, look at set of numbers, interpret and work with them.

But, when it came to shapes(quality), there was (at least before Topology) almost no clue. Equivalence, dimension, possible operations, and ... needed to be studied and defined. Informally speaking, to me, Topology is "science" of manipulating shapes. Hopefully, maybe one day, in the same vast way we operate on numbers.

Some call Topology knowledge of understanding shape of nature whether leaf formations, DNA entanglements, or quantum fields. (There is a course focusing on Topology and Biology...)

Adam Frank: Interesting, from this point of view it makes Topology sound like it shares something deeply in common with Mathematical Logic as a means of formalizing something that, for a long time, felt un-formalize-able.

And I remember reading articles in Philosophy journals which discussed topological analyses of formal logical systems and I had NO idea how that was possible, even though I have a good grasp of Logic and a (very) weak grasp of Topology. But I do know one manageable (for me) example of how they're related, in "the smallest" non-Hausdorff topology, the Sierpinski space. The elements can be thought of as true and false, and if X is any topology and U a subset and chi_U the characteristic function then chi_U is continuous just in case U is open. And I sort of remember this being interesting for something like computability or decidability or something like that. Anyway, I'm definitely glad to be making all these connections in my mind with all these other things.