Problem Set 1

Let $X$ be a set and let $d$ be the discrete metric on $X$, i.e.

$d(x,y) = \begin{cases} 0 & \text{if } x = y \\ 1 & \text{if } x \neq y \end{cases}$

(a) Fix $x \in X$. For every $r > 0$, describe what the open ball $B(x, r)$ is (with respect to the metric $d$).

(b) Recall the definition of the metric topology $\mathcal{T}_d$ induced by $d$. Which subsets $\Omega \subseteq X$ belong to $\mathcal{T}_d$?

(c) Describe all clopen sets in the topology $\mathcal{T}_d$.

Let $X$ be a nonempty set. Assume that for every $x \in X$ we have a family $\mathcal{U}_x$ of subsets of $X$ satisfying:

(1) $V \in \mathcal{U}_x \implies x \in V$

(2) $V \in \mathcal{U}_x$ and $V \subseteq W \implies W \in \mathcal{U}_x$

(3) $V_1, V_2 \in \mathcal{U}_x \implies V_1 \cap V_2 \in \mathcal{U}_x$

(4) For every $V \in \mathcal{U}_x$ there exists $W \in \mathcal{U}_x$ such that $V \in \mathcal{U}_y$ for every $y \in W$.

(a) Let $\mathcal{T} := \{\Omega \subseteq X : \Omega \in \mathcal{U}_x \text{ for every } x \in \Omega\}$. Verify that $\mathcal{T}$ is a topology.

(b1) If $U \subseteq X$ is a neighbourhood of $x$ in $\mathcal{T}$, then $U \in \mathcal{U}_x$.

(b2) If $U \in \mathcal{U}_x$, then $U$ is a neighbourhood of $x$ in $\mathcal{T}$.

Let $(X, \mathcal{T})$ be a Hausdorff topological space. Let $A \subseteq X$ and $x_0 \in X$. Prove that $x_0$ is a cluster point of $A$ if and only if for every neighbourhood $V$ of $x_0$, the intersection $A \cap V$ is infinite.

Let $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$ with $-\infty < x < +\infty$ for all $x \in \mathbb{R}$. For $x \in \mathbb{R}$, $r > 0$, $B(x, r) := (x - r, x + r)$; for $r > 0$, $B(+\infty, r) := (1/r, +\infty) \cup \{+\infty\}$ and $B(-\infty, r) := \{-\infty\} \cup (-\infty, -1/r)$. Define

$\overline{\mathcal{E}} := \{\Omega \subseteq \overline{\mathbb{R}} : \forall x \in \Omega, \exists \rho > 0 \text{ with } B(x, \rho) \subseteq \Omega\}.$

(a) Show that for every $x \in \overline{\mathbb{R}}$ and $0 < r_1 < r_2$, $B(x, r_1) \subseteq B(x, r_2)$. Deduce that $\overline{\mathcal{E}}$ is a topology on $\overline{\mathbb{R}}$.

(b) Given $A \subseteq \mathbb{R}$, show $+\infty$ is a cluster point of $A$ iff $\sup A = +\infty$.

(c) Let $A \subseteq \mathbb{R}$ with $\sup A = +\infty$, $f : A \to \mathbb{R}$, and $\lambda \in \mathbb{R}$. Prove $\lim_{x \to \infty} f(x) = \lambda$ iff $\forall \varepsilon > 0, \exists M > 0$ such that $|f(x) - \lambda| < \varepsilon$ for all $x \in A$ with $x > M$.

Consider $f : \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$.

(a) Domain discrete, codomain Euclidean. Is $\lim_{x \to 2} f(x) = 4$? Is $f$ continuous?

(b) Domain Euclidean, codomain trivial. Find all $\lambda$ with $\lim_{x \to 2} f(x) = \lambda$. Is $f$ continuous?

(c) Domain Euclidean, codomain $\mathcal{T}_{\text{lsc}} = \{\emptyset, \mathbb{R}\} \cup \{(a, +\infty) : a \in \mathbb{R}\}$ (lower semicontinuous topology). Find all $\lambda$ with $\lim_{x \to 2} f(x) = \lambda$. Is $f$ continuous?

(d) Domain and codomain Euclidean. Find all $\lambda$ with $\lim_{x \to 2} f(x) = \lambda$. Is $f$ continuous?

Let $X = \mathbb{R}$ and $\mathcal{Z} = \{\emptyset\} \cup \{\Omega \subseteq X : X \setminus \Omega \text{ is finite}\}$.

(a) Verify $\mathcal{Z}$ is a topology.

(b) For each of $A = \{\sqrt{2}\}$, $B = \{-1, 0, 2\}$, $C = [1, \infty)$, $D = \mathbb{Q}$, find the derived set, interior, closure, and boundary.