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?

Leave your thoughts in the comments below!

Recommended Books for Real Analysis

Click on the image to buy.

https://m.media-amazon.com/images/I/51KgfU+RvgL._SL160_.jpg
Introduction to Real Analysis by Bartle and Sherbert (4e)
https://m.media-amazon.com/images/I/31rJbDUVh7L._SL160_.jpg
Principles of Mathematical Analysis by Rudin
https://m.media-amazon.com/images/I/21xmoHk2GNL._SL160_.jpg
Real Analysis by Kumaresan
https://m.media-amazon.com/images/I/41icXR1neqL._SL160_.jpg
Introduction to Topology and Modern Analysis by Simmons
https://m.media-amazon.com/images/I/418FX7j0koL._SL160_.jpg
Topology of Metric Spaces

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