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January 7th, 2018, 09:32 AM   #1
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From: vienna

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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
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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.
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