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The definition of an open set S is that there exists, for any point x element of S, an open ball with center x, all of whose points belong to S.

But if S is the empty set, then it contains no point. How can we say that the above is valid if there is no point around which to build an open ball?

I can understand why the empty set is closed. Since any closed set contains all its accumulation points and since the empty set has no such accumulation point, the set of all accumulation points is empty and is therefore contained (an equal!) to the empty set.

But in the case of openness, we are faced with a P => Q where P is not true. In such case, we cannot say anything about Q, so to me, the question of whether the empty set is open or not has no answer...

I'd appreciate your mathematical wisdom on this one...

PS. After more thought, I have come to a possible "explanation". Since any statement is either true or false and since the statement that the empty set is open clearly is not false (as in not falsifiable), then it must be true. (Of course, I could also say that it is not true, so it must be false...)

But if S is the empty set, then it contains no point. How can we say that the above is valid if there is no point around which to build an open ball?

I can understand why the empty set is closed. Since any closed set contains all its accumulation points and since the empty set has no such accumulation point, the set of all accumulation points is empty and is therefore contained (an equal!) to the empty set.

But in the case of openness, we are faced with a P => Q where P is not true. In such case, we cannot say anything about Q, so to me, the question of whether the empty set is open or not has no answer...

I'd appreciate your mathematical wisdom on this one...

PS. After more thought, I have come to a possible "explanation". Since any statement is either true or false and since the statement that the empty set is open clearly is not false (as in not falsifiable), then it must be true. (Of course, I could also say that it is not true, so it must be false...)

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