Math 440, Fall 2014, Assignment 11

The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

- Saint Augustine

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

1. Connected (if your definition includes the word "disconnected," please define that as well).
2. Connected component.
3. Path.
4. Pathwise connected.

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

1. Theorem characterizing connectedness (i.e. giving two other conditions equivalent to connectedness).
2. Theorem on sufficient conditions for connectedness (Theorem 26.7(a) and (b)).
3. Theorem on products of connected spaces.
4. Theorem on connectedness of the closure.
5. Theorem on the closure of a connected component (Theorem 26.12).
6. Theorem on forward images of connected sets.
7. Intermediate Value Theorem.

Solve the following problems:

1. Problems 26E (hint: what happens when one point is removed from each space?) and 26H.

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