
theorem Th24:
  for T being non empty TopSpace, p being Element of T for N being
  net of T st p in Lim N for S being Subset of T st S = rng the mapping of N
  holds p in Cl S
proof
  let T be non empty TopSpace, p be Element of T, N be net of T such that
A1: p in Lim N;
  let S be Subset of T;
  assume S = rng the mapping of N;
  then
A2: N is_eventually_in S by Th8;
  for G being Subset of T st G is open holds p in G implies S meets G
  proof
    let G be Subset of T such that
A3: G is open and
A4: p in G;
    G = Int G by A3,TOPS_1:23;
    then reconsider V = G as a_neighborhood of p by A4,CONNSP_2:def 1;
    N is_eventually_in V by A1,YELLOW_6:def 15;
    hence thesis by A2,YELLOW_6:17;
  end;
  hence thesis by PRE_TOPC:def 7;
end;
