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 be continuous functions. Assume that Y is Hausdorff and that there is a dense subset D of X such that f(x)=g(x) for all x?D. Prove that f(x)=g(x) for all 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.

Grading Rubric

Written communication is a key piece of modern mathematics. The Discussions as well as many of the homework problems ask you to develop an argument and write it clearly. In addition to the Discussions and homework, you are required to write two formal proofs as writing assignments. These assignments are evaluated according to their validity, readability, and fluency. The definitions for those concepts are given here.

Validity – Validity corresponds to the validity of your arguments. It addresses the extent to which your method is appropriate, your calculations are correct, and your deductions follow the rules of logic.

Readability – If your written work is not readable it cannot be assessed. Since the ability to communicate Mathematics is a focal point for this class, special attention will be paid to the readability of your work.

Fluency – Mathematics is a concise and precise language, and I wish to enhance your fluency. Therefore, part of every assessment will focus on your ability to incorporate correct, established notation and terminology into your written work.

Evaluation criteria

Descriptive adjectives

Scoring

Validity

logical arguments, deductive reasoning, proper conclusions, complete reasoning

43%

Readability

organization, presentation, format, clarity, effectiveness

35%

Fluency

proper notation, proper terminology, appropriate definitions, conciseness

22%

Theorem:

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