00:00 Introduction

00:37 Epsilon ball

01:35 Notions

06:24 Examples

Q1: What is the correct definition for the open $\varepsilon$-ball $B_\varepsilon(x)$ in a metric space $(X,d)$?

A1: ${ y \in X \mid d(x,y) \neq 0 }$

A2: ${ y \in X \mid d(x,y) > \varepsilon }$

A3: ${ y \in X \mid d(x,y) < \varepsilon }$

A4: ${ y \in X \mid d(x,y) \leq \varepsilon }$

Q2: Let $(X,d)$ be a metric space. Is there an open $\varepsilon$-ball $B_\varepsilon(x)$ with $x \in X$ and $\varepsilon > 0$ which is empty?

A1: No!

A2: Yes!

Q3: Let $X = (1,5]$ and $d(x,y) = |x-y|$. Which of the following sets is open?

A1: $(1,5]$

A2: $(1,4]$

A3: $[2,4)$

A4: $[2,4]$

Q4: Let $X = (1,5]$ and $d(x,y) = |x-y|$. Which of the following sets is not closed?

A1: $(1,5]$

A2: $(1,4]$

A3: $[2,4)$

A4: $[2,4]$