Compactness for Metrics and Compactification

due by Monday, Nov 1, 2021

This assignment is based on material in lectures 18 and 19.

  1. Consider the cover of $X = [0, 1]$ by sets of the form $X\cap (a, a + \frac{1}{3})$ for $a\in \mathbb{R}$. Find a Lesbegue number for this cover (and prove that it is a Lesbesgue number).

  2. Let $X=[0, 1]\times[0, 1]\subset\mathbb{R}^2$. For which positive real numbers $d> 0$ is there a constant $C = C_d > 0$ such that for every $\varepsilon\in (0, 1)$ there is an $\varepsilon$-net of size at most $C_d(\frac{1}{\varepsilon})^d$ in $X?$ Prove your answer.

  3. Let $X$ be a topological space and let $X^*$ be its one-point compactification. Prove or disprove the following.

    1. If $X$ is compact then $X^*$ is homeomorphic to $X$.
    2. If $X$ is compact then $X^*$ is not connected.
    3. If $X$ is connected then $X^*$ is connected.
  4. Give an example of a topological space $X$ that is not locally compact and prove that it is not locally compact (you may wish to consider the SNCF metric).