Product topology, closed subset, Hausdorff

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Homework Statement



Let (X,\tau_X) and (Y,\tau_Y) be topological spaces, and let f : X \to Y be continuous. Let Y be Hausdorff, and prove that the graph of f i.e. \graph(f) := \{ (x,f(x)) | x \in X \} is a closed subset of X \times Y.

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The Attempt at a Solution



Which property of closed set should I use to prove this? Should I assume a sequence inside the graph set converging to some (x,y) \in X and then somehow show that this limit point belongs to the graph? Or should I prove that the complement of the graph set is not open? I don't know how to finish the proof with either approach. Please give me some hint.
 
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The idea here is to find (x,y) not on the graph such that every neighbourhood misses the graph. To do this, pick (x,y) not on the graph, so it is different than (x,f(x)) and separate them by neighbourhoods. You must also use continuity here.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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