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nate808 said:

it is unmathematical and meaningless in the english language to say something "has no distance between it"

look at the definitions anyway (whatever those may be for you).

Well, yes, that is if you do not require A and B to be open in S. Both statements are equivalent though.Maxos said:It is incorrect:

[tex]S \subseteq \mathbb{R}^n[/tex] is connected [tex]\leftrightharpoons \exists A, B \subset \mathbb{R}^n : A \cup B = S , A \cap \overline{B} = \varnothing , B \cap \overline{A} = \varnothing[/tex]

(Recall that A open in S ([tex]\subset \mathbb{R}^n[/tex]) means that there exist an open [tex]U \subset \mathbb{R}^n[/tex] such that [tex]U \cap S = A[/tex].)

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