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).

More Topologies and Homeomorphisms

due by Monday, Sep 20, 2021

This assignment is based on material in lectures 9, 10 and 11.

  1. Let $X$ be a set. What is the initial topology on $X$ so that all functions $f: X \to \{0, 1\}$ are continuous? Prove your result.

  2. Prove or disprove the following.

    1. Every isometric embedding is continuous.
    2. The composition of isometric embeddings is an isometric embedding.

  3. Prove or disprove that there exists a continuous bijection from $X$ to $Y$ for the following pairs of spaces.

    1. $X=(0, 1]$ and $Y= S^1$.
    2. $X = (0, 1)\cup [2, 3)$ and $Y= (1, 2)$.

  4. Prove or disprove that the following spaces are homeomorphic to $\mathbb{R^2}$.

    1. $\{(x, y)\in \mathbb{R^2}: x^2 + 2y^2 < 1 \}$.
    2. $\{(x, y)\in \mathbb{R^2}: x^2 - 2y^2 < 1 \}$.

  5. Prove that the order topology on a finite set $S$ with respect to any order is the discrete topology on $S$.

  6. Given spaces $X$ and associated covers $\Gamma$, prove or disprove in each case that $\Gamma$ is a fundamental cover of $X$.

    1. $X = [0, 2]$ and $\Gamma = \{[0, 1], [1, 2]\}$.
    2. $X = [0, 2]$ and $\Gamma = \{[0, 1], (1, 2]\}$.
    3. $X =\mathbb{R}$ and $\Gamma=\{\mathbb{Q}, \mathbb{R}\setminus \mathbb{Q}\}$.