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. 
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 welldefined (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. 

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