How Cheenta works to ensure student success?
Explore the Back-Story

TIFR 2014 Problem 28 Solution - Continuous Functions from Discrete Space


TIFR 2014 Problem 28 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta


Problem:


Let X be a topological space such that every function f: X \to \mathbb{R} is continuous. Then

A. X has the discrete topology.

B. X has the indiscrete topology.

C. X is compact.

D. X is not connected.


Discussion:


We know that if Y is a discrete space then any function g: Y \to Z is continuous.

Option A asks whether the converse to this is true in the case that Z= \mathbb{R}.

To prove/disprove whether X has the discrete topology or not it is enough to prove whether every singleton set is open or not.

If we can show that for every x\in X there exists a function f_x :X \to \mathbb{R} such that f_x^{-1} (-1,1) = {x} then we are done. Because we are given that f_x if exists must be continuous, and since (-1,1) is open in \mathbb{R} we will have the inverse image of it open in X, so x will be open in X.

Now, this target is easy to handle. We define for each x\in X

f_x (x) = 0 and f_x (y) =2 for y \neq x.

This f_x satisfies our desired property. So X is discrete.

Taking X= \mathbb{Z} (for example) shows that X  does not need to be indiscrete nor does it have to be compact.

Taking X= {0} shows that X may be connected. Of course if X has cardinality more than 1, it is not connected.


Helpdesk

  • What is this topic: Real Analysis
  • What are some of the associated concept: Continuity, Discrete Space
  • Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert


TIFR 2014 Problem 28 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta


Problem:


Let X be a topological space such that every function f: X \to \mathbb{R} is continuous. Then

A. X has the discrete topology.

B. X has the indiscrete topology.

C. X is compact.

D. X is not connected.


Discussion:


We know that if Y is a discrete space then any function g: Y \to Z is continuous.

Option A asks whether the converse to this is true in the case that Z= \mathbb{R}.

To prove/disprove whether X has the discrete topology or not it is enough to prove whether every singleton set is open or not.

If we can show that for every x\in X there exists a function f_x :X \to \mathbb{R} such that f_x^{-1} (-1,1) = {x} then we are done. Because we are given that f_x if exists must be continuous, and since (-1,1) is open in \mathbb{R} we will have the inverse image of it open in X, so x will be open in X.

Now, this target is easy to handle. We define for each x\in X

f_x (x) = 0 and f_x (y) =2 for y \neq x.

This f_x satisfies our desired property. So X is discrete.

Taking X= \mathbb{Z} (for example) shows that X  does not need to be indiscrete nor does it have to be compact.

Taking X= {0} shows that X may be connected. Of course if X has cardinality more than 1, it is not connected.


Helpdesk

  • What is this topic: Real Analysis
  • What are some of the associated concept: Continuity, Discrete Space
  • Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Menu
Trial
Whatsapp
magic-wandrockethighlight