reserve a, r, s for Real;

theorem Th16:
  for T being non empty TopSpace st ex B being Basis of T st for X
  being Subset of T st X in B holds X is connected holds T is locally_connected
proof
  let T be non empty TopSpace;
  given B being Basis of T such that
A1: for X being Subset of T st X in B holds X is connected;
  let x be Point of T;
  let U be Subset of T such that
A2: x in Int U;
  Int U in the topology of T & the topology of T c= UniCl B by CANTOR_1:def 2
,PRE_TOPC:def 2;
  then consider Y being Subset-Family of T such that
A3: Y c= B and
A4: Int U = union Y by CANTOR_1:def 1;
  consider V being set such that
A5: x in V and
A6: V in Y by A2,A4,TARSKI:def 4;
  reconsider V as Subset of T by A6;
  take V;
  B c= the topology of T & V in B by A3,A6,TOPS_2:64;
  then V is open by PRE_TOPC:def 2;
  hence x in Int V by A5,TOPS_1:23;
  thus V is connected by A1,A3,A6;
A7: Int U c= U by TOPS_1:16;
  V c= union Y by A6,ZFMISC_1:74;
  hence thesis by A4,A7;
end;
