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 January 7th, 2018, 09:32 AM #1 Newbie   Joined: Dec 2017 From: vienna Posts: 12 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: 733 Thanks: 247 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

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