ayetolu_samuel
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Hi Can someone help with this questions.

1. Let ( X, τ) be a topological and ( Y, d) a metric space. If f; g : X Y are continuous, show that the set { x1X ; f(x) = g(x)} is closed.

2. Consider the interval [ 0, 1] with the Euclidean metric and A = [ 0, 1 ] Q with the inherited metric.xhibit, and proof, a continuous map f : A R ( where R has the standard metric) such that the set { x1X ; f(x) = g(x)} is closed

3. If ( X, τ ) and ( Y, σ) are Hausdorff topological spaces, then show that X x Y with the product topology is also Hausdorff

Thanks.
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ayetolu_samuel
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OR Any resources on how to solve above questions will be appreciated.
thanks
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Gregorius
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(Original post by ayetolu_samuel)
Hi Can someone help with this questions.

1. Let ( X, τ) be a topological and ( Y, d) a metric space. If f; g : X Y are continuous, show that the set { x1X ; f(x) = g(x)} is closed.

2. Consider the interval [ 0, 1] with the Euclidean metric and A = [ 0, 1 ] Q with the inherited metric.xhibit, and proof, a continuous map f : A R ( where R has the standard metric) such that the set { x1X ; f(x) = g(x)} is closed

3. If ( X, τ ) and ( Y, σ) are Hausdorff topological spaces, then show that X x Y with the product topology is also Hausdorff

Thanks.
Let me get you started with the first one. You have two maps from X to Y. How might you use these construct a map from X to YxY? Now think about the diagonal set (y,y) in YxY. How can you use this to answer the question?
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Gregorius
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(Original post by ayetolu_samuel)
3. If ( X, τ ) and ( Y, σ) are Hausdorff topological spaces, then show that X x Y with the product topology is also Hausdorff
This is standard book-work: https://proofwiki.org/wiki/Product_o...s_is_Hausdorff
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ayetolu_samuel
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f:X------------> Y
for Instance I know that if p belong to X it gives element of Y
p ∈ X
F:X --> Y
p ---> f(p).
p can be map to Y...
Just is the little i know about this course sir, can share resources or explain further for me sir.
Thanks for helping







(Original post by Gregorius)
Let me get you started with the first one. You have two maps from X to Y. How might you use these construct a map from X to YxY? Now think about the diagonal set (y,y) in YxY. How can you use this to answer the question?
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ayetolu_samuel
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Thanks
(Original post by Gregorius)
This is standard book-work: https://proofwiki.org/wiki/Product_o...s_is_Hausdorff
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Gregorius
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(Original post by ayetolu_samuel)
2. Consider the interval [ 0, 1] with the Euclidean metric and A = [ 0, 1 ] Q with the inherited metric.xhibit, and proof, a continuous map f : A R ( where R has the standard metric) such that the set { x1X ; f(x) = g(x)} is closed
I can't make much sense of this question; there appears to be something missing. I think it's asking you to exhibit a function f with particular properties, and prove it has these properties. But the question makes reference to an undefined function g. It looks like the question has been constructed by cutting and pasting in a wrong bit of another question!
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Gregorius
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(Original post by ayetolu_samuel)
f:X------------> Y
for Instance I know that if p belong to X it gives element of Y
p ∈ X
F:X --> Y
p ---> f(p).
p can be map to Y...
Just is the little i know about this course sir, can share resources or explain further for me sir.
Thanks for helping
You have f, g: X-->Y, so construct a function H: X-->YxY by H(x) = (f(x), g(x)). Let D = {(y, y)} in YxY (the diagonal subset). Then your subset is the inverse image of D under H. What do you need to show to complete the question?
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ayetolu_samuel
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2. Consider the interval [ 0, 1] with the Euclidean metric and A = [ 0, 1 ]∩ Q with the inherited metric.xhibit, and proof, a continuous map f : A → R ( where R has the standard metric) such that the set { x1 ∈ X ; f(x) = g(x)} is closed
(Original post by Gregorius)
I can't make much sense of this question; there appears to be something missing. I think it's asking you to exhibit a function f with particular properties, and prove it has these properties. But the question makes reference to an undefined function g. It looks like the question has been constructed by cutting and pasting in a wrong bit of another question!
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ayetolu_samuel
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Sir, can you provide a link to books/resources that can explain this topic thoroughly because the school material am using is not helping me at all......
Thanks sir

(Original post by Gregorius)
You have f, g: X-->Y, so construct a function H: X-->YxY by H(x) = (f(x), g(x)). Let D = {(y, y)} in YxY (the diagonal subset). Then your subset is the inverse image of D under H. What do you need to show to complete the question?
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Gregorius
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(Original post by ayetolu_samuel)
Sir, can you provide a link to books/resources that can explain this topic thoroughly because the school material am using is not helping me at all......
Thanks sir
I'll happily look out some links for you to lecture notes on these topics. However, can you tell me what course you're studying? This will help me choose the right level for you.
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