[4] [5] 1. Introduction X P (X) X S(X) X N R (X, c) c c ∗ : P (X) −→ P (X) c ∗ (A) = {x ∈ X|x n → x {x n } ∈ S(A)} A ⊂ X [5] c ∗ (X, c) 1.1. Lemma. ([5]) (X, c) c (1) c ∗ (φ) = φ (2) A ⊂ c ∗ (A) ⊂ c(A) A ⊂ X (3) A ⊂ B ⊂ X c ∗ (A) ⊂ c ∗ (B) (4) c ∗ (A ∪ B) = c ∗ (A) ∪ c ∗ (B) A, B ⊂ X c ∗ X (X, c) c 1.2. Definition. X (1) X [1] A X A {x n } ∈ S(A) {x n } x x ∈ A (2) X [1] A X x ∈ c(A) {x n } ⊂ A {x n } x 1 - 2000 Mathematics Subject Classification. 54A20, 54D35, 54D55, 54D80, 55E25. - Keywords. sequential, Fr´echet, weakly first countable, countable tightness, closure operator, sequential closure op erator, countable closure operator. - Received 10/01/2006, in revised form 25/02/2006. (3) X X [1] A X A c(C) ⊂ A C A (4) X [5] x ∈ X {B(x, n)|n ∈ N} X (a) x ∈ B(x, n + 1) ⊂ B(x, n) n ∈ N (b) U X X x ∈ U n ∈ N B(x, n) ⊂ U (5) X [1] X [1] [5] (1) ⇒ ⇒ ⇒ (2) ⇔ [5] (1) (C ∗ ) A X c ∗ (A) = c(A) (2) (C ∗∗ ) A X x ∈ c(A) \ A B A x ∈ c(B) c(B) \ {x} ⊂ A (3) (C ∗∗∗ ) A X x ∈ c(A)\A B A x ∈ c(B) c(B)\{x} ⊂ A [4] [5] C C c(C) C c ∗ = c [4] 1.3. Lemma. (1) ([5], Theorem 2) (C ∗ ) (2) ([5], Theorem 4) (C ∗∗∗ ) (C ∗ ) (3) ([5], Corollary 6) (C ∗∗ ) c ∗ [4] [5] [4] [5] 2. The main results (X, c) c c ∗ : P (X) −→ P (X) A ⊂ X c ∗ (A) = {x ∈ X|x ∈ c(C) C A }, (X, c) 2.1. Theorem. (X, c) c (1) c ∗ (φ) = φ (2) A ⊂ B c ∗ (A) ⊂ c ∗ (B) A, B ⊂ X (3) c ∗ (A ∪ B) = c ∗ (A) ∪ c ∗ (B) A, B ⊂ X (4) c ∗ (A) = c(A) A X (5) A ⊂ c ∗ (A) ⊂ c ∗ (A) ⊂ c(A) A ⊂ X (6) c ∗ (A) = (c ∗ ) ∗ (A) A ⊂ X (c ∗ ) ∗ (A) = c ∗ (c ∗ (A)) Proof. A ⊂ A ∪ B B ⊂ A ∪ B c ∗ (A) ∪ c ∗ (B) ⊂ c ∗ (A ∪ B) x ∈ c ∗ (A ∪ B) x ∈ c(C) C A ∪ B c(C) = c((C ∩ A) ∪ (C ∩ B )) = c(C ∩ A) ∪ c(C ∩ B), x ∈ c(C ∩ A) x ∈ c(C ∩ B) C ∩ A C ∩ B A B x ∈ c ∗ (A) ∪ c ∗ (B) c ∗ (A) ⊂ (c ∗ ) ∗ (A) x ∈ (c ∗ ) ∗ (A) x ∈ c(C) C c ∗ (A) C = {x i : i ∈ N} ⊂ c ∗ (A) i ∈ N x i ∈ c(C i ) C i A  i∈N C i A C ⊂  i∈N c(C i ) ⊂ c   i∈N C i  x ∈ c(C) ⊂ c   i∈N C i   i∈N C i A (c ∗ ) ∗ (A) ⊂ c ∗ (A) c ∗ (A) = (c ∗ ) ∗ (A)  2.2. Remark. (1) A X c ∗ (A) = c ∗ (A) = c(A) (2) (C ∗ ) (C ∗ )  (C ∗ )  A X c ∗ (A) = c ∗ (A) (3) 2.3. Theorem. (X, c) c (X, c) c ∗ (A) = c(A) A ⊂ X Proof. (X, c) A ⊂ X c(A) ⊂ c ∗ (A) A ⊂ c ∗ (A) c ∗ (A) C c ∗ (A) x ∈ c(C) y ∈ C ⊂ c ∗ (A) C y A y ∈ c(C y ) D =  y∈C C y D A U x U ∩ C = φ y ∈ U y ∈ C y ∈ c(C y ) U ∩ C y = φ U ∩ D = φ x ∈ c(D) c(C) ⊂ c ∗ (A) (X, c) c ∗ (A) c ∗ (A) = c(A) A ⊂ X B ⊂ X c(C) ⊂ B C B c ∗ (B) ⊂ B c(B) ⊂ B B (X, c)  2.4. Corollary. X (1) X (2) X (3) c ∗ (A) = c ∗ (A) = c(A) A ⊂ X 2.5. Corollary. (X, c) c (X, c ∗ ) Proof. (X, c ∗ ) c ∗ c ∗ (c ∗ (A)) = c ∗ (A) A ∈ P (X) (X, c ∗ )  2.6. Corollary. (X, c) (C ∗ ) Proof. (X, c) (C ∗ ) c ∗ (A) = c(A) A X c ∗ (A) = c ∗ (A) A X c ∗ (A) ⊂ c ∗ (A) x ∈ c ∗ (A) C A x ∈ c(C) (X, c) (C ∗ ) c(C) = c ∗ (C) ⊂ c ∗ (A) x ∈ c ∗ (A) c ∗ (A) = c ∗ (A) c ∗ (A) = c ∗ (A) = c(A) A X X  2.7. Corollary. X (1) X (2) X (3) X (C ∗ ) (4) A X c ∗ (A) = c ∗ (A) = c(A) Proof. ⇔ ⇒ ⇒ ⇒  X k-space A ∈ P (X) A ∩ K K K X [7] k [1] k 2.8. Example. X = {0}∪( ∞  i=1 X i ) X i =  1 i  ∪( ∞  j=i 2  1 i + 1 j  ) i = 1, 2, · · · X {B(x)|x ∈ X} x = 1 i + 1 j B(x) = {x} x = 1 i B(x)  1 i  ∪ ( ∞  j=k  1 i + 1 j  ), k = i 2 , i 2 + 1, · · · x = 0 B(0) X X i  1 i + 1 j  X i Y = X \  1, 1 2 , · · ·  k [1] Y Y [4] [5] 2.9. Lemma. (1) ([5], Corollary 5) (C ∗∗∗ ) (2) ([4], Theorem 2.10) (C ∗∗ ) 2.10. Remark. (1) (2) [5] (3) [5] X = {a, b, c} {φ, X, {a}} X (C ∗ ) (C ∗∗∗ ) 2.11. Example. Y k Y Y (C ∗∗ ) A Y {i|A ∩ X i } 0 /∈ A [1] c(A) = A ∪ {0} c(A) \ {0} ⊂ A (C ∗∗ ) Y (C ∗∗ ) Y (C ∗∗∗ ) Y Y (C ∗ ) 2.12. Example. X = R Y = (R\N)∪{∞} x ∈ X f(x) =  x x ∈ R \ N, ∞ x ∈ N. Y {A ⊂ Y |f −1 (A) X}. [1] (1) A Y A ∞ ∈ A U U X N ⊂ U A = (U \ N) ∪ {∞}. (2) A Y A ∞ /∈ A A = B B X B ∩ N = φ (3) Y (4) Y (C ∗∗ ) Y (C ∗∗ ) Y (C ∗∗ ) (C ∗∗ ), (C ∗∗ ) A X X x ∈ c(A) \ A B A x ∈ c ∗ (B) c ∗ (B) \ {x} ⊂ A X (C ∗∗ ) (C ∗∗ ) 2.13. Theorem. X A X B A c ∗ (B) ⊂ A Proof. X A X B A c(B) ⊂ A B c(B) = c ∗ (B) B ⊂ A c ∗ (B) ⊂ A A ⊂ X c(C) ⊂ A C A A B A c ∗ (B) ⊂ A C ⊂ B ⊂ A c(C) ⊂ A A X  2.14. Corollary. (C ∗∗ ) Proof. X (C ∗∗ ) A X x ∈ c(A) \ A B A x ∈ c ∗ (B) c ∗ (B) \ {x} ⊂ A c ∗ (B) ⊂ A X  2.15. Remark. X = {a, b, c} {φ, {a}, X} (C ∗ ) (C ∗∗ ) [5] X (C ∗∗ ) References [1] R. Engelking, General topology, PWN-Polish Scientific Publishers, Warszawa 1977. [2] S. P. Franklin, Spaces in which sequences suffice, Fund. Math., 57 (1965), 108 - 115. [3] S. P. Franklin, Spaces in which sequences suffice II, Fund. Math., 61 (1967), 51 - 56. [4] W. C. Hong, Notes on Fr´echet, Internat. J. Math. & Math. Sci., 22 (3) (1999), 659 - 665. [5] W. C. Hong, On sequential spaces and a related class of spaces, Kyungpook Math. J., 40 (2000), 149 - 155. [6] B. Skorulski, First countable, Sequential. and Fr´echet spaces, J. of Formalized Math., Vol. 10 (2003), 1 - 5. [7] Y. Tanaka, Theory of k-networks II, Q & A in Topology, Vol. 19 (2001), 27 - 46. [4] [5] . 665. [5] W. C. Hong, On sequential spaces and a related class of spaces, Kyungpook Math. J., 40 (2000), 149 - 155. [6] B. Skorulski, First countable, Sequential. and Fr´echet spaces, J. of Formalized. Franklin, Spaces in which sequences suffice, Fund. Math., 57 (1965), 108 - 115. [3] S. P. Franklin, Spaces in which sequences suffice II, Fund. Math., 61 (1967), 51 - 56. [4] W. C. Hong, Notes on Fr´echet,. 54D80, 55E25. - Keywords. sequential, Fr´echet, weakly first countable, countable tightness, closure operator, sequential closure op erator, countable closure operator. - Received 10/01/2006, in revised