CSIR-NET: Mathematical Sciences, GATE Mathematics, Real Analysis, Topology, Uncategorized

Examples and Counterexamples: Real Line with Discrete Metric/Topology

Posted by jessepfrancis on Mar 8, 2020Apr 11, 2020

🛈 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:

Every function is continuous in $\mathbb{R}_d$ .
Every set is open subset of $\mathbb{R}_d$ .
Every set is closed in $\mathbb{R}_d$ .
$\mathbb{R}_d$ is not complete.

Examples of interest

$f(x)=$ greatest integer smaller than or equal to x is not continuous in $\mathbb{R}$ with usual metric, but is continuous in $\mathbb{R}_d$ .
$\{a\},a\in\mathbb{R}_d$ is not open in $\mathbb{R}$ , but it is open in $\mathbb{R}_d$ . In fact, every set is open in $\mathbb{R}_d$
$\{a\},a\in\mathbb{R}_d$ is closed in $\mathbb{R}$ , and in $\mathbb{R}_d$ .

Challenges

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

Leave your thoughts in the comments below!

Recommended Books for Real Analysis

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

Published by jessepfrancis

Christian. Artist. Mathematician. Programmer. Teacher. Visit https://mathematicos.in/aboutjesse/ for my full profile. View all posts by jessepfrancis

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