# Topology

Watch
Announcements
#1
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.
0
#2

OR Any resources on how to solve above questions will be appreciated.
thanks
Last edited by ayetolu_samuel; 2 years ago
0
2 years ago
#3
(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?
0
2 years ago
#4
(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
0
#5
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?
0
#6
Thanks
(Original post by Gregorius)
This is standard book-work: https://proofwiki.org/wiki/Product_o...s_is_Hausdorff
0
2 years ago
#7
(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!
0
2 years ago
#8
(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?
0
#9
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!
0
#10
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?
0
2 years ago
#11
(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.
0
X

new posts Back
to top
Latest
My Feed

### Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

### See more of what you like onThe Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

### Poll

Join the discussion

#### If you haven't confirmed your firm and insurance choices yet, why is that?

I don't want to decide until I've received all my offers (44)
40.74%
I am waiting until the deadline in case anything in my life changes (21)
19.44%
I am waiting until the deadline in case something in the world changes (ie. pandemic-related) (6)
5.56%
I am waiting until I can see the unis in person (9)
8.33%
I still have more questions before I made my decision (9)
8.33%
No reason, just haven't entered it yet (8)
7.41%
Something else (let us know in the thread!) (11)
10.19%