1. Topological Spaces (18/8/2022).
  2. Bases (25/8/2022).
  3. Metric Spaces (8/9/2022).
  4. Subsets and Continuity (15/9/2022).

Subsets and Continuity

due by Thursday, Sep 15, 2022

  1. Let $A$ be the subset of the Cantor set $X$ (recall this is defined as $\{0, 1\}^\mathbb{N}$ with an ultra-metric) with $A$ consisting of sequences $\{a_n\}$ such that $a_n= 0$ for all but finitely many $n$. Prove or disprove the following.

    1. $A$ is open in $X$.
    2. $A$ is closed in $X$.
    3. $A$ intersects every open set in $X$.

  2. Prove or disprove the following.

    1. A subspace of a space with discrete topology has discrete topology.
    2. A subspace of a space with indiscrete topology has indiscrete topology.
    3. A subspace of a space with cofinite topology has cofinite topology.

  3. Let $Y$ be a topological space such that for all topological spaces $X$, every map $f: X\to Y$ is continuous. Prove that $Y$ has the indiscrete topology.