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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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, 1) ∈ Y and 2 ∈ X to 1 ∈ Y and the
rest can be mapped via identity...