Hello all of the questions must be hand written with full shown work..thanks Also we are using the...

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Hello all of the questions must be hand written with full shown work..thanks Also we are using the book "Introduction to Topology: Pure and Applied" by Colin Adams. I am going to attach the first four chapters of the book. Since our proffesor wants us to use the book only for the answers. thanks please try to use the theorems from the book .

David · Accepted Answer

1
6. Suppose that X is Hausdorff. We would like to show it’s image under 4 is closed. We can do that by showing
that it’s complement 4(X)c is open. 4(X) consists of points with equal coordinates so 4(X)c consists of
points (x, y) with x and y distinct.
For any x, y ∈ 4(X)c, the Hausdorff condition gives us disjoint open U, V ⊂ X with x ∈ U, y ∈ V . Then
U × V is a basis element containing (x, y). U and V have no points in common, so U × V contains nothing in
the image of the diagonal map: U × V contained in 4(X)c. So 4(X)c is open making 4(X) closed.
Now lets supppose 4(X) is closed. Then 4(X)c is open. Given any (x, y) ∈ 4(X)c, there is a basis element
U ×B with (x, y) ∈ U × V ⊂ 4(X)c. The basis element containing (x, y) gives us open disjoint U, V with
x ∈ U, y ∈ V . Hence X is Hausdorff.
7.(a) One example of a bijection f : X → Y would be to send (0, 1) ∈ X to (0,

Hello all of the questions must be hand written with full shown work..thanks Also we are using the book "Introduction to Topology: Pure and Applied" by Colin Adams. I am going to attach the first four...

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