Math 440, Fall 2014, Assignment 7

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

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

1. Directed set.
2. Net.
3. Eventually in (a set $U$).
4. Frequently in (a set $U$).
5. Convergence (of a net to a point).
6. Cluster point (of a net).
7. Subnet.

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

1. Theorem characterizing closures in terms of nets (Theorem 11.7).
2. Theorem characterizing continuous maps in terms of nets (Theorem 11.8).
3. Theorem characterizing cluster points of a net in terms of subnets (Theorem 11.5).

Solve the following problems:

1. Problems 11A(1) and 11B(3).

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