The Pre T ½ Spaces (The New Further Results) Dr. Abdul Salaam
... sets. In this way, Maki, Umehara and Noiri [5] define the concept of pregeneralized closed sets of a topological space taking help of the preopen sets. In the present paper, we continue to give some characterizations for pre T ½ spaces. Also, we introduce the equivalence between a pre T ½ space and ...
... sets. In this way, Maki, Umehara and Noiri [5] define the concept of pregeneralized closed sets of a topological space taking help of the preopen sets. In the present paper, we continue to give some characterizations for pre T ½ spaces. Also, we introduce the equivalence between a pre T ½ space and ...
point set topology - University of Chicago Math Department
... Lemma 3.5. Let X be first countable. Then x ∈ Ā if and only if there is a sequence {xn } ⊂ A such that {xn } → x. Using Lemma 2.2 this leads to the promised characterization of continuity. Proposition 3.6. Let f : X −→ Y be a function, where X is first countable and Y is any space. Then f is contin ...
... Lemma 3.5. Let X be first countable. Then x ∈ Ā if and only if there is a sequence {xn } ⊂ A such that {xn } → x. Using Lemma 2.2 this leads to the promised characterization of continuity. Proposition 3.6. Let f : X −→ Y be a function, where X is first countable and Y is any space. Then f is contin ...
TIETZE AND URYSOHN 1. Urysohn and Tietze Theorem 1. (Tietze
... map F : X → [0, 1] with F |A = f . (iii) For all disjoint closed subsets B1 , B2 of X, there is a Urysohn function: a continuous function f : X → [0, 1] with B1 ⊂ f −1 (0) and B2 ⊂ f −1 (1). Proof. (i) =⇒ (ii): We directly follow an argument of M. Mandelkern [Ma93]. Let A ⊂ X be a closed subset of a ...
... map F : X → [0, 1] with F |A = f . (iii) For all disjoint closed subsets B1 , B2 of X, there is a Urysohn function: a continuous function f : X → [0, 1] with B1 ⊂ f −1 (0) and B2 ⊂ f −1 (1). Proof. (i) =⇒ (ii): We directly follow an argument of M. Mandelkern [Ma93]. Let A ⊂ X be a closed subset of a ...