Math 440, Fall 2014, Assignment 4

No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....

- Morris Kline, Mathematics in Western Culture

Carefully define the following terms, then give one example and one non-example of each:

1. Subspace topology (on a subset $A$ of a topological space $X$).
2. Function continuous at a given point.
3. Continuous function.
4. Homeomorphism (from one space top another).
5. Homeomorphic (spaces).
6. Topological property.

Carefully state the following theorems (you do not need to prove them):

1. Theorem characterizing continuous functions (Theorem 7.2).
2. Theorem on composition of continuous functions.
3. Theorem on restriction of continuous functions (Theorem 7.5).
4. Theorem concerning continuity of functions defined on unions (Theorem 7.6).
5. Theorem concerning restriction of codomains (Theorem 7.7).

Solve the following problems:

1. Problems 6C, 7A, 7C, and 7G.

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