Puzzle: Three Connected Sets

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

True or False: Let A and B be two connected subsets of a metric space. If B is a set such that A ⊆ B ⊆ C, then B is connected.
Wrong. Taking counterexamples, let A = {1}, B = {1} U {2}, C = [1,2]. Clearly, the hypothesis is satisfied, but B is not connected.
To prove, you need a proof. To disprove, you need a counterexample. Leave your proof below in comments!

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

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

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