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What should be a good example to 2 sets say X, Y $\subset$ $\mathbb R^n$ and these are separated strictly but their closures $\bar X$ and $\bar Y$ can't be strictly separated ?

Definition of Strict Separation: There are stronger notions of separation. The hyperplane [p = $\alpha$] strictly separates A and B if A and B are in disjoint open half spaces, that is, A $\subset$ [p > $\alpha$] and B $\subset$ [p < $\alpha$] (or vice versa).

If you give and explain the example it will be great for me to understand the fact.


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What do you mean by strictly separated? Do you mean open sets that properly contain them? – Clayton Jan 7 '13 at 21:34
Yes, two open sets and a hyperplane separating them. That is the definition of strict separation. – DreamLighter Jan 7 '13 at 21:38
up vote 0 down vote accepted

Assuming $A$ and $B$ are strictly separated means that there exist open sets $U$ and $V$ such that $A\subsetneq U$, $B\subsetneq V$ and $U\cap V=\varnothing$.

Let $A=[0,1]\times[0,\frac{1}{2})$ and $B=[0,1]\times(\frac{1}{2},1]$; this is the unit square with the line $y=1/2$ deleted. Then this makes it clear that both sets can be strictly separated, say $$U=(-\frac{1}{2},\frac{3}{2})\times (-\frac{1}{2},\frac{1}{2})$$ and $V$ defined analogously, even the closure of one of them prevents them from being able to be separated.

It is also clear how to generalize this to $n$ dimensions.

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