Homework 10, Real Analysis

Due November 2

Problem 1

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


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


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

Problem 2

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


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

Problem 3

State whether the following are true or false.

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


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

Problem 4

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

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


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