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I think this would only work if you know x. X cannot be in the formula for finding x, thus it must be manipulated such that x is not in the formula, correct?

You take the finest topology that guarantees the canonical projection (that maps a point to its equivalence class) is continuous. It can be constructed explicitly as {𝑉 ⊆ 𝑋/~ | π^(−1)(𝑉) ∈ τ} where π is the canonical projection and τ is the topology of the original space.

I know, this known as the final topology of pi. But this doesn't really cut anything does it? How would you cut the reals (\R) into (-infty,0) and [0,infty) using an equivalence relation? Edit: and have the cut be \R/~.

@@meisterschiumpf9759 Quotient spaces don’t “cut” spaces, they “glue” them by collapsing together points that the equivalence relation binds. I suppose that in your first comment you referred to the usage of the word “cut” in the introduction. It’s not needed to turn a square into a torus. But if you want to do it the other way around (turn a torus into a square), you’d have to start from your torus and then find another topological space with an equivalence relation whose quotient space is homeomorphic to your torus. That would be a cut in that sense. Another strategy would be to duplicate your torus, define your cut as a continuous path, remove a bit of surface on the first torus on one side of the cut, remove a bit of surface on the second torus on the other side of the same cut, define an equivalence relation that binds together two corresponding points that remain in both toruses and then quotient those out (this construction is called adjunction space).

@@pierrecolin6376 yes, i did refer to the word cut. I just wanted to know what he means by cut since i never heard of a way to do so. I just read wikipedia on adjunction space. There it says it is used to glue things, which also is very logical to me if i look at its construction. But i still want to cut... Though sure, nice to know. Also if we remove anyway it should just be enough to remove the two lines where the space was glued or not? I like this, thank you. Edit: btw for a torus X let ~={(a,a):a in X}. Then ofc X/~=X so i think your first approach cannot be formalized and therefore is not that interesting, right?

@@meisterschiumpf9759 Whether removing just the cut line or not is enough depends on what you want to happen at the edge along that cut. When you cut your torus back into a square, do you want the edges to be included or not? If not, you can get away with just taking the subset topology on the complementary of the cutting line. If so, I don’t think there’s an easy way. For a torus you could take your square as a subset of a euclidean space and take its closure to get the edges back, but I don’t know how well this generalizes.

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