reserve GX for TopSpace;
reserve A, B, C for Subset of GX;
reserve TS for TopStruct;
reserve K, K1, L, L1 for Subset of TS;
reserve GX for non empty TopSpace;
reserve A, C for Subset of GX;
reserve x for Point of GX;

theorem Th44:
  for T being TopSpace, X being set holds
  X is connected Subset of T iff X is connected Subset of the TopStruct of T
proof
  let T be TopSpace, X be set;
  thus X is connected Subset of T implies X is connected Subset of the
  TopStruct of T
  proof
    assume
A1: X is connected Subset of T;
    then reconsider X as Subset of T;
    reconsider Y=X as Subset of the TopStruct of T;
    the TopStruct of T|X = (the TopStruct of T)|Y by PRE_TOPC:36;
    then (the TopStruct of T)|Y is connected by A1,Def3;
    hence thesis by Def3;
  end;
  assume
A2: X is connected Subset of the TopStruct of T;
  then reconsider X as Subset of the TopStruct of T;
  reconsider Y=X as Subset of T;
  the TopStruct of T|Y = (the TopStruct of T)|X by PRE_TOPC:36;
  then T|Y is connected by A2,Def3;
  hence thesis by Def3;
end;
