Topology

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.

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

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?

This is standard book-work: https://proofwiki.org/wiki/Product_of_Hausdorff_Factor_Spaces_is_Hausdorff

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

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

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!

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?

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