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happysauce
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Homework Statement
Let f:X→Y we a continuous map. Let
Γ(x)={(x,f(x))∈X×Y}
a) Show that if X is connected, then Γ is connected.
b) Show that if Y is hausdorrf then Γ is a closed subset of X×Y.
Homework Equations
thm: if X is connected the image of a continuous map f:X→Y is connected.
thm: The product space of two connected spaces is connected.
thm: X is hausdorrf iff the diagonal is closed (?)
def: A space is hausdorrf if each pair of distinct points have neighborhoods that are disjoint.
The Attempt at a Solution
a) Since f:X→Y is continuous, the image is going to be connected. Therefore image(f(X))[itex]\subset[/itex] Y is connected. Also given two connected spaces the product space is also connected, therefore X×image(f(X)) is connected. Therefore Γ is connected? This seemed too easy so I am not sure if I left a huge hole here...
b) Y is hausdorrf. Therefore given any points y1≠y2 we can find a disjoint neighborhoods U and V around y1 and y2 respectfully. I also know that the diagonal, D, is closed, {(y,y)|y is in Y}. Not sure what to do next. Maybe consider the complement and try to do a proof similar to the diagonal proof? The complement of D would be open. I don't think Y being hausdorrf implies that f(x1) and f(x2) have disjoint neighborhoods so I'm not sure where to go next.
Edit: *Being Hausdorrf implies that sequences converge to at most one point. *Does this mean that f(x_n) converges to f(x) in this case? *
I don't want any flat out answers, I just want to know what I should be looking for.
Also new here to the physics forums, if my write up is awful please tell me :)
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