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We know that, indiscrete topology is the smallest topology. It has $2$ elements (they are the empty set and whole set). Suppose the given set is the empty set, then how can I define a topology on that set? Is it possible?

miradulo
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Avinash N
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    There is only one possible topology on the empty set---the set of the empty set. This is a discrete, Hausdorff, second-countable, separable, connected, totally disconnected, path-connected, compact topology. – Batominovski Jul 09 '15 at 18:31
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    There is only one topology on the empty set, it is ${\emptyset}$. In this case, the empty set and the whole set are the same. – Michael Burr Jul 09 '15 at 18:32

2 Answers2

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If the underlying $X$ of a topological space is even allowed to be empty under our system of definitions, then $\{ \emptyset \}$ is the only such topology. Note that this is the same as the indiscrete topology $\{ \emptyset,X \}$ since $X=\emptyset$. It is also the same as the discrete topology $\mathcal{P}(X)$.

Ian
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  • yes correct. But i found some new ideas from some text books. They says "Topology on a set has cardinality atleast 2" – Avinash N Jul 09 '15 at 18:35
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    @AvinashN If the set is nonempty then that is true. There is really not much lost in defining the underlying set of a topological space to be nonempty, so they might be doing that. – Ian Jul 09 '15 at 18:36
  • {∅} this is also a discrete topology... – Avinash N Jul 09 '15 at 18:36
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    Correct (this is in my last sentence, which got edited in). – Ian Jul 09 '15 at 18:36
  • What is correct definition for topology? – Avinash N Jul 09 '15 at 18:38
  • @AvinashN Really?... – AlexR Jul 09 '15 at 18:40
  • You can permit the empty topological space or not; this is pretty much the decision of the individual mathematician. If you do permit it, you get a bunch of pretty ugly counterexamples. – Ian Jul 09 '15 at 18:40
  • In the definition of Group, vector space, etc we seen that the given set is non empty. Thats why i am confused with topology on the empty set. – Avinash N Jul 09 '15 at 18:45
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    @AlexR So, since Wikipedia allows $X=\emptyset$, then definitions where $X=\emptyset$ is not allowed, are wrong definitions? I think the right response to AvinashN is that definitions are neither 'correct' nor 'incorrect'. In think AvinashN's last comment may be directed to you. – BCLC Sep 11 '18 at 07:16
  • Am I right in thinking that this is the only indiscrete topology that is Hausdorff? (as the requirement for a Hausdorf topology is vacuously true) – Marcus Junius Brutus Mar 07 '24 at 17:50
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The empty space on $\pi$-Base: https://topology.pi-base.org/spaces/S000163

Several of the standard properties hold vacuously. In particular any statement that must hold for all points, nonempty open sets, etc., will be true because these things don't exist in the first place.

Steven Clontz
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