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I've been studying topology this term and it really got me interested. But sometimes in math I feel that we are just taught things one by one, without really talking about why we do it that way. So I tend to ask those question: why we use this, why not that, what if this?

Back to topology. The topology, I was taught was a lot about continous maps, homeomorphisms, homotopy - in other words, everything was continous. In some way, it makes sense. A lot of things in the real world obey continuity (for example knots - this is not probably the best example, but still). It is, I think, clear, that omitting the continuity in e.g. Knot Theory, would make the theory somewhat trivial. It is very inefficient to undo knots using scissors, but every knot could be undone that way.

But then, we can use scissors. I mean, where else, than in math we should try to work with different premises, axioms,.. I see, that as topology allows the flexibility, allowing cutting might make the theory trivial (everything would be possible, in some sense). So maybe omitting the flexibility and allowing cutting and gluing?

Continuity is very reasonable premise, but than again, isn't cutting also? Continuity is probably easier to work with, but allowing cutting would tied our hands?

Question

Is there such type of topology, that I've described? Isn't it simply metric spaces?

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    $\begingroup$ To me the option is indeed there: the reverse of the quotient operation (which we think of as gluing) is cutting. $\endgroup$ – Ian May 30 '15 at 14:42
  • $\begingroup$ Yes, I've thought of that too, should have mentioned that, thanks! What about homeomorphisms, homotopy? Allowing cutting would identify e.g. the circle and segment. Does this premise simplify classifying objects? E.g. A surface would be identified with some kind of space filling curve, that is with a line? $\endgroup$ – quapka May 30 '15 at 14:51
  • $\begingroup$ You're right. You are being taught things one by one without full motivations and explanations. The reason is that these motivations and explanations sometimes require a lot of previous knowledge (about mathematical history, the development of the topic (in this case topology), and sometimes the mathematical familiarity with the mathematics of the time of development). This is not something that every teacher has under their belt, and even more rare are the undergrad students who posses all these things. Let me qualify this by saying that some motivation can, and should be given, to students. $\endgroup$ – Asaf Karagila May 30 '15 at 14:51
  • $\begingroup$ @AsafKaragila I understand that, and I've experienced myself, that it can be very hard to describe and motivate something in math to someone else. I am not judging my teachers, merely noting that. I do not expect to have everything served to me, it just took me some time to realize, that sometimes we don't do certain things that way, because we must, but because we simply chose to.. $\endgroup$ – quapka May 30 '15 at 15:09
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"Continuity" and "cutting" are not in any way incompatible notions. I would say that you're conflating the mathematical notion of continuity (that is, continuity of a function) with the common-language meaning of the word. For example, most people would say that this space $X$: $$\huge \mathbf{-}\qquad \mathbf{-}$$ is "not continuous", or perhaps that it "has been cut" from the space $Y$ $$\huge \mathbf{-\!\!\!\!\!\!-}$$

While $X$ is disconnected (Wikipedia link), it is nevertheless a perfectly ordinary topological space, and it makes just as much sense to talk about the continuity (or non-continuity) of a given function $f:X\to Z$ for any other space $Z$.

Ian puts it quite well in a comment above, that the operation of "cutting" a space into pieces can be implemented as simply observing that your current space can be obtained by "gluing", i.e. quotienting, those pieces together. Observe that a "gluing" map (such as the one "gluing" $X$ back together into $Y$ above) is a continuous function in the ordinary sense; no new mathematics is necessary to realize the intuitive notion of "cutting".

Another example of how "cutting" is already a perfectly well-behaved concept within the standard world of topology, would be the subject of surgery theory (Wikipedia link).

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  • $\begingroup$ Thanks, I understand the duality of cutting and gluing. I am sorry, for creating the confusion. I am interested also in the discontinous maps, sth like discontinous homotopy. So, circle would be indetified with some segment, although they are not homeomorphic (you would need to cut the circle, or glue the segment for that, but that is not allowed in homemorphism). $\endgroup$ – quapka May 30 '15 at 15:16

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