**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*→*Y*f, g: X→Y*Y* is Hausdorff and that there is a dense subset *D* of *X*such that *f(x)=g(x)* for all *x*∈*D*x∈D *f*(*x*)=*g*(*x*)f(x)=g(x) *x*∈*X*x∈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.

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