reserve a, r, s for Real;

theorem
  for S, T being non empty TopSpace st S,T are_homeomorphic & S is
  locally_connected holds T is locally_connected
proof
  let S, T be non empty TopSpace;
  given f being Function of S,T such that
A1: f is being_homeomorphism;
  assume
A2: S is locally_connected;
  now
    let A be non empty Subset of T, B being Subset of T;
    assume A is open & B is_a_component_of A;
    then
A3: f"A is open & f"B is_a_component_of f"A by A1,Th10,TOPGRP_1:26;
    rng f = [#]T by A1;
    then f"A is non empty by RELAT_1:139;
    then f"B is open by A2,A3,CONNSP_2:18;
    hence B is open by A1,TOPGRP_1:26;
  end;
  hence thesis by CONNSP_2:18;
end;
