Some Surveys On Hyperspaces 2~x

The paper discuss compactness, locally compactness, countable compactness, sequential compactness of hyperspace 2X and separation of hyperspace 2X , the main results obtained as follows:1. Let X be a topological space,Then the space X is compact if and only if hyperspace 2X is compact.2. Let X be a T₂ space .Then the following statements are equivalent: (1) X is compact. (2) 2X is compact. (3) 2^X is Lindelo f. (4) 2X is paracompact. (5) 2^X is metacompact. (6) 2X is meta-Lindelo f. (7) 2^X is locally meta-Lindelo f. (8) There is < U 1 , U 2, Un>? 2X such that i j for each (?)i≤n, X∈< U₁ , U₂,…U_n> and < U₁ , U₂,…U_n> is a meta- meta-Lindelo f.3. If X be a regular space. Then the following statements are equivalent:(1)2^X is compact.(2)2^X is locally compact.(3) X is compact.4. Let X be a T₂ space. Then hyperspace 2^X is locally sequential compact if and only if 2^X is sequential compact.5. If X be a closed separable inherit and closed countable characteristic of regular space, and hyperspace 2^X is countable compact, then hyperspace 2^X is sequential compact. 6. Let X be a T2 space. Then hyperspace 2X is locally countable compact if and only if 2X is countable compact.Corollary 1: If X be a regular space. P0( X )= { A | A ? X且A≠?}, then the following statements are equivalent:(1) P0( X ) is compact.(2) P0( X ) is locally compact.(3) X is compact.Corollary 2: Let X be a T2 space. If hyperspace 2X is locally sequential compact, then X is sequential compact.