The Connectedness of the Graph of a Continuous Map in Hausdorrf Spaces

[happysauce]

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))\subset 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 :)

 

[happysauce]

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))\subset 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 :)

I might have it. consider y≠f(x) then (y,f(x)) is in the complement of Γ. Take y in U and f(x) in V. Since Y is hausdorrf x and f(x) have disjoint nighborhoods in Y. Suppose U\cap V ≠∅ then there exists x=f(x), which is a contradiction, therefore given any ordered pair (y,f(x)) the neighborhood UxV is open in the complement of the graph? Therefore the graph is closed?

 

[jbunniii]

a) Since f:X→Y is continuous, the image is going to be connected. Therefore image(f(X))\subset 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?

How did you conclude that? It's certainly not true that an arbitrary subset of a connected set is connected.

Try working with the definition. Assume \Gamma(X) is disconnected. Then \Gamma(X) is the disjoint union of two open sets U and V. What can you say about the inverse images, \Gamma^{-1}(U) and \Gamma^{-1}(V), of these sets?

 

[happysauce]

How did you conclude that? It's certainly not true that an arbitrary subset of a connected set is connected.

Try working with the definition. Assume \Gamma(X) is disconnected. Then \Gamma(X) is the disjoint union of two open sets U and V. What can you say about the inverse images, \Gamma^{-1}(U) and \Gamma^{-1}(V), of these sets?

I concluded this with a theorem from my book. "The image of a connected space under a continuous map is connected." So I thought maybe given X is connected the image of f was connected, and I just threw in the subset of Y because every point in the graph is an element in XxY.

And to answer your question.
Γ−1(U) and Γ−1(V) are disjoint and open in X?

 

[happysauce]

I concluded this with a theorem from my book. "The image of a connected space under a continuous map is connected." So I thought maybe given X is connected the image of f was connected, and I just threw in the subset of Y because every point in the graph is an element in XxY.

And to answer your question.
Γ−1(U) and Γ−1(V) are disjoint and open in X?

Actually since this is the inverse image of f, then wouldn't that mean that Γ−1(U) and Γ−1(V) are disjoint and open in X and their union is the whole space X? Contradicting the fact that X was connected?