Proof of The Equivalence of Statements

Assignment
Part 1: Write a complete proof of the equivalence of statements (1) and (2) in Theorem 4.11 of Croom.

Part 2: Next, write a complete proof of the following statement:

Let f, g: X→Yf, g: X→YY is Hausdorff and that there is a dense subset D of Xsuch that f(x)=g(x) for all x∈Dx∈D f(x)=g(x)f(x)=g(x) x∈Xx∈X

As with Writing Assignment #1, use a solid organizational structure:
– State the hypotheses
– State the conclusions
– Clearly and precisely prove the conclusions from the hypotheses
– Results presented earlier in the text may be used and must be clearly documented
A few notes about format: use Microsoft Word; use Equation Editor for all mathematical symbols, e.g. x ∈ X or Cl(A) ⋂ Cl(X-A); and use the fonts Cambria and Cambria Math in size 11 so your typed work is the same font as your equations.
Course and Learning Objectives
This Writing Assignment supports the following Course and Learning objectives:
CO2: Use homeomorphisms and show two spaces are topologically equivalent.
LO2: Define relevant continuous functions and homeomorphisms