Research report: "the probability measure functor preserving some topological properties that" ppt

P k (X) X P k (X) 1. Introduction and notions cs cs ∗ X X µ : X → [0, 1] (1) µ = {x ∈ X : µ(x) > 0} (2)  x∈ µ µ(x) = 1 k ∈ N P k (X) X k µ ∈ P k (X) µ = q  i=1 m i δ x i q  k δ x (y) δ x (y) =  0 y = x 1 y = x, m i = µ(x i ) > 0, q  i=1 m i = 1 m i = µ(x i ) µ x i P k (X) µ ∈ P k (X) µ =  q i=1 m i δ x i µ O(µ, U 1 , U 2 , . . . , U q , ε) = {µ ∈ P k (X) : µ = q+1  i=1 µ i , µ i ⊂ U i , | µ i  −m i |< ε, i = 1, 2, , q +1, m q+1 = 0}, 1 ε > 0, U 1 , U 2 , , U q x 1 , x 2 , . . . , x q U q+1 = X\  q i=1 U i  µ i =  x∈ µ T U i µ(x) µ i i = 1, 2, , q + 1  µ i = 1. B X {O(µ, U 1 , , U q , ε) : µ ∈ P k (X); U i ∈ B, i = 1, 2, , q; q  k, ε > 0} P k (X) Fedorchuk topology µ = q  i=1 m i δ x i ∈ P k (X), µ n = q  i=1 m n i δ x n i , q  k {µ n } µ o ε > 0 N > 0 n ≥ N x n i ∈ U i , i = 1, . . . , q |m n i − m o i | < ε, i = 1, . . . , q X x ∈ P ⊂ X P x {x n } x P k ∈ N x n ∈ P n ≥ k X P X (1) P X x ∈ X U x P ∈ P x ∈ P ⊂ U. (2) P cs X {x n } x ∈ X U x m ∈ N P ∈ P {x}  {x n : n ≥ m} ⊂ P ⊂ U (3) P cs ∗ X {x n } ⊂ X x ∈ X U x {x n i : i ∈ N} {x n } P ∈ P {x}  {x n i : i ∈ N} ⊂ P ⊂ U (4) P wcs ∗ X {x n } x ∈ X U x {x n i } {x n } P ∈ P {x n i : i ∈ N} ⊂ P ⊂ U P =  x∈X P x X P x ∈ X P x x P 1 P 2 ∈ P x P ∈ P x P ⊂ P 1 ∩ P 2 (1) P X G ⊂ X G X x ∈ G P ∈ P x P ⊂ G P x x (2) P sn X P x x x ∈ X P x sn x 2. The main results X µ = q  i=1 m i δ x i ∈ P k (X), q  k P X P i ∈ P, i = 1, 2, . . . q ε > 0 P ∗ µ P ∗ µ = P ∗ (µ, P 1 , P 2 , . . . , P q , ε) = =  µ = q+1  i=1 µ i : µ i ⊂ P i , | µ i  −m i |< ε, i = 1, 2, . . . , q; P q+1 = X \ q  i=1 P i , m q+1 = 0; P i  P j = φ with i = j, and  µ i =  x i ∈ µ T P i µ(x i ),  µ q+1 < ε  . P ∗ µ = {P ∗ (µ, P 1 , P 2 , . . . , P q , ε) : P i ∈ P, i = 1, . . . , q, ε > 0}, P ∗ =  µ∈P k (X) P ∗ µ . P ∗ P k (X) P X P ∗ P k (X) µ P k (X) µ = q  i=1 m i δ x i ∈ P k (X), q  k O(µ, U 1 , U 2 , . . . , U q , ε) µ P ∗ µ ∈ P ∗ µ ∈ P ∗ µ ⊂ O(µ, U 1 , . . . , U q , ε). U i ∩ U j = ∅ i = j P i = 1, . . . , q P i ∈ P x i ∈ P i ⊂ U i i = 1, . . . , q P ∗ µ = P ∗ (µ, P 1 , . . . , P q , ε) µ ∈ P ∗ µ ⊂ O(µ, U 1 , , U q , ε) P ∗ P k (X) P =  x∈X P x X P (1) P x (2) P 1 , P 2 ∈ P x P ∈ P x P ⊂ P 1  P 2 P ∗ =  µ∈P k (X) P ∗ µ P ∗ 1 , P ∗ 2 ∈ P ∗ µ P ∗ 1 = P ∗ (µ, P 1 1 , . . . , P 1 q , ε 1 ) where ε 1 > 0 P ∗ 2 = P ∗ (µ, P 2 1 , . . . , P 2 q , ε 2 ), where ε 2 > 0 µ = q  i=1 m i δ x i ∈ P k (X), q  k P 1 i  P 1 j = φ i = j P 1 i ∈ P x i , i = 1, 2, . . . , q P 2 i  P 2 j = φ i = j P 2 i ∈ P x i , i = 1, 2, . . . , q P 1 i ∩ P 2 i = ∅ i = 1, . . . , q P P 3 i ∈ P x i P 3 i ⊂ P 1 i  P 2 i i = 1, . . . , q ε < min{ε 1 , ε 2 } P ∗ 3 = P ∗ (µ, P 3 1 , . . . , P 3 q , ε) P ∗ 3 ⊂ P ∗ 1  P ∗ 2 . P sn X P ∗ sn P k (X) µ ∈ P k (X) P ∗ = P ∗ (µ, P 1 , . . . , P q , ε) ∈ P ∗ µ µ µ = q  i=1 m i δ x i m i = µ(x i ) > 0 δ x i i = 1, . . . , q {µ n } µ P k (X) µ n = q+1  i=1 µ n i µ n −→ µ ε > 0 n i ∈ N µ n i ⊂ P i , | µ n i  −m i |< ε, i = 1, 2, . . . , q;  µ n q+1 < ε n ≥ n i n o = max{n 1 , n 2 , . . . , n q } n ≥ n o µ n ∈ P ∗ (µ, P 1 , P 2 , . . . , P q , ε) P ∗ µ P cs ∗ X P ∗ cs ∗ P k (X) {µ n } µ P k (X) O(µ, U 1 , . . . , U q , ε) µ µ =  q i=1 m i δ x i µ n =  q+1 i=1 µ n i q  k µ n −→ µ m ∈ N µ n ∈ O(µ, U 1 , . . . , U q , ε), for every n ≥ m. n ≥ m µ n i ⊂ U i , | µ n i  −m i |< ε, i = 1, 2, . . . , q  µ n q+1 < ε i = 1, . . . , q n ≥ m x n i ∈ µ n i µ n −→ µ {x n i } x i i = 1, . . . , q P cs ∗ X x n i −→ x i i = 1, . . . , q i = 1, . . . , q U i x i P i ∈ P {x n j i } {x n i } {x n j i }  {x i } ⊂ P i ⊂ U i P ∗ = P ∗ (µ, P 1 , P 2 , . . . , P q , ε) n j µ n j =  q+1 i=1 µ n j i µ n j ∈ P k (X) {µ n j }  {µ} ⊂ P ∗ = P ∗ (µ, P 1 , P 2 , . . . , P q , ε) ⊂ O(µ, U 1 , U 2 , . . . , U q , ε). P cs X P ∗ cs P k (X) {µ n } µ P k (X) O(µ, U 1 , . . . , U q , ε) µ µ =  q i=1 m i δ x i µ n =  q+1 i=1 µ n i q  k µ n −→ µ m ∈ N µ n ∈ O(µ, U 1 , . . . , U q , ε), for every n ≥ m. n ≥ m µ n i ⊂ U i , | µ n i  −m i |< ε, i = 1, 2, . . . , q  µ n q+1 < ε i = 1, . . . , q n ≥ m x n i ∈ µ n i µ n −→ µ {x n i } x i i = 1, . . . , q P cs X x n i −→ x i i = 1, . . . , q i = 1, . . . , q U i P i ∈ P m i ∈ N {x i }  {x n i : n ≥ m i } ⊂ P i ⊂ U i . m = max{m 1 , m 2 , . . . , m q } P ∗ = P ∗ (µ, P 1 , P 2 , . . . , P q , ε) {µ}  {µ n : n ≥ m} ⊂ P ∗ ⊂ O(µ, U 1 , U 2 , . . . , U q , ε). P ∗ = P ∗ (µ, P 1 , P 2 , . . . , P q , ε) ∈ P ∗ . P wcs ∗ X P ∗ wcs ∗ P k (X) [1] V. V. Fedorchuk, Soviet Math. Dokl. , 1986, pp. 1329 -1333. [2] Ta Khac Cu, VNU, Journal of science, , 2003, pp. 22 - 33. [3] G. Gruenhage, E. Michael and Y. Tanaka, Pacific J. Math., , 1984, pp. 303 - 332. [4] Y. Tanaka, k , Topology Proc., , 1987, pp. 327 - 349. [5] S. Liu and C. Liu, cs Topology Appl., , 1996, pp. 51 - 60. [6] Y.Ge, Publication de L’institute Mathema- tique, Nouvelle serie, , 2003, pp. 121 - 128. [7] Y. Ikeda and Y. Tanaka, k Topology Proc., , 1993, pp. 107 - 132. P k (X) X P k (X)