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.
  1. Can you find an example of compact set in \mathbb{R}_d?
  2. Can you characterize convergent sequences in \mathbb{R}_d?

Leave your thoughts in the comments below!

Recommended Books for Real Analysis

Introduction to Real Analysis by Bartle and Sherbert (4e)
Principles of Mathematical Analysis by Rudin
Real Analysis by Kumaresan
Introduction to Topology and Modern Analysis by Simmons
Topology of Metric Spaces

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