MAT 530 Topology/Geometry I
October 31, 1995
- 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
- 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.
- 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
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]