Can anyone give me an metric space that is not complete and not separable??
(I can think of the other three combinations.)
Thank you!!
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3Not is not hard. Take a space that is not complete, and one that is not separable. Paint one of them red and the other blue, and declare that the distance between blue points and red points is $1$. – André Nicolas Aug 05 '13 at 18:30
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Take a non-separable Hilbert space, and consider the subspace spanned by an orthonormal basis (that is, the subspace of finite linear combinations of the ONB elements). – Daniel Fischer Aug 05 '13 at 18:32
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Take your favourite non-separable metric space $A$, for concreteness an uncountable set with all non-zero distances equal to $1$. Let $B$ be the non-complete space the open interval $(0,1)$ with the usual distance. and your favourite non-complete metric space. We can arrange for $A$ and $B$ to be disjoint.
Let $X=A\cup B$. Define a distance function on $X$ by using the inherited distances for points that are both in $A$, or both in $B$. If $a\in A$ and $b\in B$, declare the distance from $a$ to $b$ to be $1$. Then $X$ equipped with this metric is neither separable nor complete.
André Nicolas
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If $a \in A$ and $b \in B$ and declare the distance from $a$ to $b$ to be $1$. Then, for $a_1, a_2 \in A, b\in B$, we may not be able to satisfy $d(a_1,a_2) \leq d(a_1,b) + d(a_2,b)=2$? I mean, it seems that I cannot verify the distance function defined in this way is a metric. – poormaths Aug 06 '13 at 04:57
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Indeed, thank you, I made a mistake. The thing can be fixed by altering the metric on $A$ and $B$ to bring it down below $1$. I do not have time to write that up now, so will delete. – André Nicolas Aug 06 '13 at 05:04
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