My Math Forum  

Go Back   My Math Forum > College Math Forum > Topology

Topology Topology Math Forum


Thanks Tree2Thanks
  • 2 Post By Micrm@ss
Reply
 
LinkBack Thread Tools Display Modes
January 7th, 2018, 09:32 AM   #1
Newbie
 
Joined: Dec 2017
From: vienna

Posts: 9
Thanks: 1

topology

In a metric, can one separate any two (disjoint) closed/bounded sets through open sets?

Last edited by skipjack; January 7th, 2018 at 02:14 PM.
aminea95 is offline  
 
January 7th, 2018, 03:23 PM   #2
Senior Member
 
Joined: Oct 2009

Posts: 608
Thanks: 186

Yes, every metric space is normal. This means exactly that any two disjoint closed sets $A$ and $B$ can be separated by open sets.

This is rather easy to prove too. Consider the function
$$f: (X,d)\rightarrow [0,1] : x \rightarrow \frac{d(x,A)}{d(x,A) + d(x,B)}$$
Note that the function is well-defined (why?).
The function satisfies $f(A) = \{0\}$ and $f(B) = \{1\}$. The function is also clearly continuous.
We say that $f$ functionally separates $A$ and $B$.

Now, to get your two disjoint open sets, simply consider $f^{-1}([0,1/2))$ and $f^{-1}((1/2,1])$.

Separation of bounded sets is in general not possible. Consider in $\mathbb{R}$ the disjoint bounded sets $[0,1)$ and $[1,2]$. These cannot be separated by open sets.
Thanks from aminea95 and Snair
Micrm@ss is offline  
Reply

  My Math Forum > College Math Forum > Topology

Tags
topology



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Topology johnny_01 Topology 4 November 3rd, 2012 09:41 PM
Problem on product topology/standard topology on R^2. vercammen Topology 1 October 19th, 2012 12:06 PM
Topology Artus Topology 5 September 5th, 2012 08:21 AM
discrete topology, product topology genoatopologist Topology 0 December 6th, 2008 11:09 AM
discrete topology, product topology Erdos32212 Topology 0 December 2nd, 2008 02:04 PM





Copyright © 2018 My Math Forum. All rights reserved.