Topological Spaces

due by Monday, Aug 16, 2021

This assignment is based on material in lectures 1 and 2.

  1. For sets $A$ and $B$, recall that $A\setminus B= \{a\in A: a \notin B\}$. Which of the following always equals $A\setminus (A \setminus B)$? Prove your answer.

    1. $A$
    2. $B$
    3. $A \cap B$
    4. $A\cup B$
  2. Let $X=\{1, 2\}$. What is the number of collections of subsets $\Omega\subset X$ that form a topology on $X$? Prove your answer.

  3. Prove that the cofinite topology on a set $X$ equals the discrete topology if and only if $X$ is finite.