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Is $\phi$ set forms a metric space or not ?

I think, it does not form a metric space, because, we can't specify a metric on $\phi$.

But, In many text book, it is not mention that, the set on which, we define metric should be non empty.

If I may suppose, that d is a function define on $\phi$ $ \times$ $\phi$ such that

d is constant function with range set { $0$ }. Then it must be metric on {$\phi$}.

Plz help... what is the right thingh ?

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    [Why are metric spaces non-empty?](http://math.stackexchange.com/q/45145/) might be related (you still haven't specified whether you mean the empty space or the space containing the empty set as only point).2013-01-03

2 Answers 2

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Yes. A singleton is a metric space.

To see that, note that a subspace of a metric space is a metric space, and $\{x\}$ is a subset of $\mathbb R$ whenever $x\in\mathbb R$.

Since all singletons are "essentially" the same, this means that $\{\varnothing\}$ can also be thought as a metric space.

On the other hand whether or not $\varnothing$ itself, the empty set, is a metric space is up to definition, whether or not you are allowing empty structures in your universe, or does the empty set carries no structure.

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Obviusly $(\{\emptyset\},d)$ defines a metric space, the trivial one. Or you can say vacuously. You can check that the rules of a metric space are satisfied.