Homework 9, Real Analysis

Due date: October 6

Problem 1

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

Problem 2

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 3

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.

Problem 4

1. The fixed points of a continuous $f:\mathbb B^n\to\mathbb B^n$ might not be interior.


2. The Brouwer fixed point theorem is false for the open ball.