Non-continuous topology? 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?
 A: "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).
