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  1. May the empty set be a topological space? Is the set $\{\emptyset\}$ its topology?
  2. What about the empty metric space?
Nurlan Abiev
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    Does it satisfy the axioms of a topological space? – Mark Saving Mar 16 '21 at 17:31
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    My professor defined a metric/topological space as a nonempty set equipped with ... . I must therefore thank her for having saved us from the present fruitless discussions! – Black Mar 16 '21 at 18:34
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    @Black You have been made a considerable disservice, because you'll have to make caveats on the existence of coproducts of families of spaces, the existence of products will have to be unnaturally subordinated to the axiom of choice and Tychonov theorem will be way less elegant. –  Mar 16 '21 at 18:47
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    @Black Not including the empty topological space means that its not a complete or cocomplete category. We need the empty topological space, it's very important. – Noel Lundström Mar 16 '21 at 19:40

1 Answers1

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  1. Definitely yes and yes. In point of fact, it's the initial object of the category of topological spaces. It is compact, but as far as I know it is currently a matter of debate whether it should be considered connected or not.

  2. I would say yes (like wikipedia does). It's homeomorphic to the empty topological space and it still is the initial object in all the categories I can think of putting on metric spaces (namely: the category with continuous maps, or the category of uniformly continuous maps, or the category of Lipschitz maps, or the category of non-expansive maps).