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Due November 2

Problem 1

1. If $A\subset X$ is connected, then so is its closure $\bar A$.

2. If A is connected and $A \subset B \subset \bar A$, then so is B.

3. If $A\subset \R^n$ is convex, then $\bar A$ is convex.

Problem 2

1. If $f:[0,1]\to[0,1]$ is continuous, then there exists $c\in[0,1]$ such that $f(c) = c$.

2. If $f:\mathbb S^1\to\R$ is continuous, then there exists $\xi\in\mathbb S^1$ such that $f(\xi) = f(-\xi)$.

Problem 3

State whether the following are true or false.

1. If $A,B$ are path connected, then $A\cap B$ is path connected.

2. If $A, B\subset\R^n$ are convex, then $A\cap B$ is convex.

Problem 4

Let $A\cap B\not=\emptyset$ in some metric space. State whether the following are true or false.

1. If $A,B$ are path connected, then $A\cup B$ is path connected.

2. If $A,B\subset\R^n$ are convex, then $A\cup B$ is convex.