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- Thread starter nate808
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[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]

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matt grime

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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).

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

[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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