Math 145 Practice Problems for Final Fall 2003

Review Midterm, Theorems from the book, problems from Homework.

The final will cover Sections 1.1-1.9, 2.1-2.6, 3.2, 3.4-3.7, 4.1-4.6, 5.1-5.4

1. Let $X$ and $Y$ be topological spaces, $f: X \rightarrow Y$ a function. Let $\cal G$ be a basis for the topology on $Y$. Suppose that if $O \in {\cal G}, f^{-1} (O)$ is open in $X$. Prove that $f$ is continuous. (Hint: Use Homework Problem 4(a) in Page 14.)

2. Let $A \subset X$ be a closed set of a topological space $X$. Let $B\subset A$ be a subset of $A$. Prove that $B$ is closed as a subset of $A$ iff $B$ is closed as a subset of $X$. Show that this is false if we omit the assumption that $A$ is closed.

3. Find a topological space $X$ so that $X$ and $X\times X$ are homeomorphic. (Hint: $\emptyset$ is an easy answer. Try for a more interesting one, at first using discrete topologies.)

4. Prove that the real line is homeomorphic to the open interval $(0,1)$, or in fact any open interval.

5. Show that the line and the plane are not homeomorphic.

6. Show that the $n$-sphere

$$S^n = \{x\vert x \in \mathbb{R}^{n+1}, d(x,0) =1\}$$

is connected for $n \ge 1$.

7. Let $Z$ be a connected subspace of $X$. Suppose $W$ is a subspace between $Z$ and its closure, that is

$$Z \subset W \subset \bar{Z}.$$

Prove that $W$ is connected.

8. Let $f,g$ be two continuous functions from a metric space $S$ to the real numbers. Define

$$h(x) = \max (f(x),g(x)).$$

Prove tht $h(x)$ is continuous.

9. Find all topologies on the set of three points which are Hausdorff.

10. Let $f: X \rightarrow \mathbb{R}$ be a continuous function, with $X$ compact. Prove that $f$ is bounded, i.e., there is a $K$ with $\vert f(x)\vert <K$ for all $x\in X$.