## Midterm Examination

October 31, 1995

1. Prove from the definitions: The continuous image of a connected set is connected.

• Prove from the definitions: If X is compact and Y is Hausdorff and f: X --> Y is continuous and one-to-one, then f is a homeomorphism onto its image.

• Show by an example that the hypothesis ``X compact'' is necessary.

• Prove that any uncountable subset of the plane R^2 has a limit point.

• Suppose A: R^2 --> Z is any function from the plane to the integers. For each n let S_n be the set of points x such that A(x) = n. Prove that for at least one number n the set S_n has a limit point in R^2.

2. What topological properties of the real line are used in proving the following two statements?

• The ``Intermediate Value Theorem:'' If [a,b] is a closed interval, and f a continuous function defined on [a,b], then for any c in the interval [f(a),f(b)] there is an x in [a,b] with f(x) = c.

• If f is a continuous real-valued function defined on a closed interval [a,b], then there exists x in [a,b] such that f(x) > = f(y) for any y in [a,b]