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.
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