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Akhilesh posted an Question
April 04, 2022 • 03:02 am 30 points
  • IIT JAM
  • Mathematics (MA)

1. let t and t' be two topologies on x. if t' 3 t, what does connectedness of x in one topology imply about connectedness in the other?

1. Let T and T' be two topologies on X. If T' 3 T, what does connectedness of X in one topology imply about connectedness in the other?

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  • Anonymous User best-answer

    If (X,T) is disconnected, then there exists a separation (U,V), where U and V are disjoint open non-empty subsets of (X,T) Since we are given T ⊂ T', then U and V are open non-empty subsets of (X,T'), and their union is X. Then, (U,V) is a separation of (X,T'), so (X,T') is disconnected. Conversely, if (X,T') is connected, then (X,T) is connected.

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