The curves are actually pairwise disjoint simple closed, but that wouldn't fit in the title.
I assume the answer is $n$ but I'm not sure how to prove it, or how to visualize it. Any help or a push in the right direction would be appreciated
Find the largest number of curves in $n\mathbb{T}$ such that cutting $n\mathbb{T}$ along the curves gives a single path-connected piece
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general-topology
algebraic-topology
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1$n\mathbb{T}$ is a torus with $n$ holes? – 2017-02-16
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0Yes! I've thought about it for a bit and I think the answer is $n+1$. – 2017-02-16
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0Sorry to keep posting, as I'm not very good at these things, but $n$ seems correct. For example, if $n=0$, then the statement is essentially the Jordan curve theorem – 2017-02-16
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0That's what I was thinking, but I wasn't quite sure what my book meant by "cuts". I assume it was just removing the curve, but if that's what it is then there are two distinct curves in $\mathbb{T}$ the one that turns the torus into a cylinder (by cutting "into the donut") and the other being "around the donut" – 2017-02-16
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0Again, hopefully someone better at this stuff comes along, but I think that can't be what is intended. The answer is infinite then, even if you consider curves up to homotopy or homology. – 2017-02-16
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0Few suggestions for writing/editing your question(s): Do not try to fit your question into the title, write instead something like: "Curves on surfaces". Then in the main body of the question write the detailed question itself with all the definitions. Do not expect the reader to know the definitions that your instructor uses in the class, unless they are from a textbook (and even then). Indicate what did you try, e.g. did you make a computation with $n=0$? Or $n=1$? Or $n=2$? – 2017-02-16