A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset

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Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, then[1][2]:

  • [ilmath](X,\mathcal{ J })[/ilmath] is connected if and only if the only two sets that are both open and closed in [ilmath](X,\mathcal{ J })[/ilmath] are [ilmath]X[/ilmath] itself and [ilmath]\emptyset[/ilmath]

Proof

Grade: C
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The message provided is:
See Connected_(topology)#Equivalent_definition if stuck, but it's pretty easy

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References

  1. Introduction to Topological Manifolds - John M. Lee
  2. Introduction to Topology - Bert Mendelson