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.
TRUE
Wrong. Taking counterexamples, let A = {1}, B = {1} U {2}, C = [1,2]. Clearly, the hypothesis is satisfied, but B is not connected.
FALSE
To prove, you need a proof. To disprove, you need a counterexample. Leave your proof below in comments!

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

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 )

Google photo

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

Twitter picture

You are commenting using your Twitter 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.