1. Topological Spaces (16/8/2021).
  2. Bases and Metrics (23/8/2021).
  3. Subsets and Metrics (30/8/2021).
  4. Continuity (13/9/2021).
  5. More Topologies and Homeomorphisms (20/9/2021).
  6. Connectedness and Applications (11/10/2021).
  7. Separation Properties (18/10/2021).
  8. Countability and Compactness (25/10/2021).
  9. Compactness for Metrics and Compactification (1/11/2021).
  10. Products and Metrization (15/11/2021).
  11. Baire Theorem and Quotients (22/11/2021).

Subsets and Metrics

due by Monday, Aug 30, 2021

This assignment is based on material in lectures 5 and 6.

  1. Let $X$ be a metric space, $x, y\in X$ points and $A\subset X$ a subset. Prove or disprove the following.

    1. $d(A, y) \leq d(A, x) + d(x, y)$.
    2. $d(x, y) \leq d(A, x) + d(A, y)$.

  2. Suppose we consider the Hausdorff distance on all bounded sets (i.e., not necessarily closed). Prove or disprove the following.

    1. $d_H(A, B) = 0$ if and only if $A= B$.
    2. $d_H(A, B) = d_H(B, A)$.
    3. $d_H(A, C)\leq d_H(A, B) + d_H(B, C)$.

  3. Let $A$ be the subset of the Cantor set $X$ 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 $X$.

  4. 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.