Homework 10, Real Analysis

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.

 

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