Math 440, Fall 2014, Assignment 6

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

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

1. Sequence (in a topological space).
2. Convergence (of a sequence to a point).
3. First countable (topological space).

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

1. Theorem characterizing closures in first countable spaces in terms of sequences (Theorem 10.4).
2. Example showing the failure of the preceding theorem in general topological spaces (Example 10.6(a)).

Solve the following problems:

1. Problems 10A(1,3,5,6) and 10B.

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