Examples and Counterexamples: Real Line with Discrete Metric/Topology

🛈 This materials is useful for Unit 1 in CSIR-NET.

Definition: $<\mathbb{R},d>$, where $d(x,y)=\begin{cases}0 & x=y\\ 1&x\ne y\end{cases}$

Prove that $d$ is a metric on $\mathbb{R}$, as an exercise.

We denote it as $\mathbb{R}_d$.

In terms of topological spaces, $<\mathbb{R},\tau>,\tau=\mathscr{P}(\mathbb{R})$, where $\mathscr{P}(\mathbb{R})$ is the power set of $\mathbb{R}$

Some Interesting Properties:
1. Every function is continuous in $\mathbb{R}_d$.
2. Every set is open subset of  $\mathbb{R}_d$.
3. Every set is closed in $\mathbb{R}_d$.
4. $\mathbb{R}_d$ is not complete.
Examples of interest
1. $f(x)=$ greatest integer smaller than or equal to x is not continuous in $\mathbb{R}$ with usual metric, but is continuous in $\mathbb{R}_d$.
2. $\{a\},a\in\mathbb{R}_d$ is not open in $\mathbb{R}$, but it is open in $\mathbb{R}_d$. In fact, every set is open in $\mathbb{R}_d$
3. $\{a\},a\in\mathbb{R}_d$ is closed in $\mathbb{R}$, and in $\mathbb{R}_d$.
Challenges
1. Can you find an example of compact set in $\mathbb{R}_d$?
2. Can you characterize convergent sequences in $\mathbb{R}_d$?