Abstract. In this paper, we investigate the relationships between the space X and the hyperspace C ( X ) concerning admissibility and connectedness im kleinen. The following results are obtained: Let X be a Hausdorff continuum, and let A ∈ C ( X ). (1) If for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ⊂ IntK ⊂ K ⊂ U , then C ( X ) is connected im kleinen . at A . (2) If IntA ≠ ø, then for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ⊂ IntK ⊂ K ⊂ U . (3) If X is connected im kleinen . at A , then A is admissible. (4) If A is admissible, then for any open subset U of C ( X ) containing A , there is an open subset V of X such that A ⊂ V ⊂ ∪ U . (5) If for any open subset U of C ( X ) containing A , there is a subcontinuum K of X such that A ∈ IntK ⊂ K ⊂ U and there is an open subset V of X such that A ⊂ V ⊂ ∪ IntK , then A is admissible. Let X be a Hausdorff continuum, and let 2 ^X ( C ( X ), K ( X ), C_K ( X )) the hyperspace of nonempty closed subsets(connected closed subsets, compact subsets, continua) of X with the Vietoris topology. Throughout by a continuum we mean a compact connected Hausdorff space. For a continuum X , C ( X ) is endowed with the Vietoris topology and, since X is a continuum, the hyperspace C ( X ) is also a continuum [8] . Wojdyslawsk [13] established the conditions of local connectedness between a space X and its hyperspace 2 ^X ( C ( X )). Goodykoontz [3 , 4 , 5] investigated local connectedness as a pointwise property in the hyperspace 2 ^X ( C ( X )) of metric continua. And Goodykoontz and Rhee [6] investigated the relationships between the space X and the hyperspaces concerning the properties of local compactness and local connectedness. They proved that a Hausdorff space X is connected im kleinen at x ∈ X if and only if 2 ^X ( K ( X ), C_K ( X )) is connected im kleinen at { x } and a locally compact Hausdorff space X is connected im kleinen at x ∈ X if and only if 2 ^X ( C ( X ), K ( X ), C_K ( X )) is connected im kleinen at { x }. In 2003, Makuchowski [9 , 10] investigated with respect to local connectedness at a subcontinuum of continua. The purpose of this paper is to investigate the relationships between the space X and the hyperspace C ( X ) concerning admissibility and connectedness im kleinen. For notational purposes, small letters will denote elements of X , capital letters will denote subsets of X and elements of 2 ^X , and script letters are reserved for subsets of 2 ^X . If B ⊂ 2 ^X , ∪ B = { A : A ∈ B }. If A ⊂ X , the symbol will denote the interior(closure, boundary) of the set A . Let X be a topological space. Let 2 ^X = { E ⊂ X : E is nonempty and closed}, K ( X ) = { E ∈ 2 ^X : E is compact}, C ( X ) = { E ∈ 2 ^X : E is connected}, and C_K ( X ) = K ( X ) ∩ C ( X ), and endow each with the Vietoris topology. A basis for 2 ^X consists of all elements of the form < U ₁ , U ₂ , ⋯ , U_n >= { A ∈ 2 ^X : A ∩ U_i ≠ ø for each i and where U ₁ , U ₂ , ⋯ , U_n are open sets in X . Let T ( x ) = { A ∈ C ( X ) : x ∈ A }. An element A ∈ T ( x ) is said to be admissible at x in X if, for each basic open set < U ₁ , U ₂ , ..., U_n > ∩ C ( X ) containing A , there is a neighborhood V_x of x in X such that whenever y ∈ V_x there is an element B ∈ T ( y ) such that B ∈< U ₁ , U ₂ , ..., U_n > ∩ C ( X ) [11] . The space X is said to be locally connected at x in X , if for each neighborhood U of x there is a connected neighborhood V of x such that V ⊂ U [7] . The space X is said to be connected im kleinen at x , if for each neighborhood U of x there is a component of U which contains x in its interior [7 , 9] . The space X is said to be locally connected provided that X is locally connected at each of its points. If a space X is connected im kleinen at each of its points, then X is locally connected. The space X is said to be locally arcwise connected at x , if for each neighborhood U of x there is an arcwise connected neighborhood V of x such that V ⊂ U . The space X is said to be locally arcwise connected , if X is locally arcwise connected at each of its points. The space X is said to be arcwise connected im kleinen at x , if for each neighborhood U of x there is an arcwise connected, component of U which contains x in its interior. If a space X is arcwise connected im kleinen at each of its points, then X is locally arcwise connected. A continuum X is said to be connected im kleinen at a subcontinuum A , if for each open subset U of X containing A , there is a subcontinuum K such that A ⊂ IntK ⊂ K ⊂ U [10] . A continuum X is said to be locally connected at a subcontinuum A , if for each open subset U of X containing A , there is an open connected subset V such that A ⊂ V ⊂ U [1] . Obviously, if a subcontinuum is degenerate, then the notion of connectedness im kleinen(local connectedness) at a subcontinuum is the same as the notion of connectedness im kleinen(local connectedness) at a point. Note that if X is connected im kleinen(locally connected) at each point of A , then X is connected im kleinen(locally connected) at a subcontinuum A , but not conversely, Result 1.1 ( [12] ). (Boundary Bumpping Theorem) Let X be a Hausdorff continuum, and let A ∈ C ( X ). Then for each open set U in X containing A, the component C_A of containing A intersects Bd ( U ). Theorem 2.1. Let X be a Hausdorff continuum, and let A ∈ C ( X ). If for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ⊂ IntK ⊂ K ⊂ U, then C ( X ) is connected im kleinen at A . Proof. Let U =< U ₁ , ⋯ , U_n > ∩ C ( X ) be an open subset of C ( X ) containing A . Then is an open subset of X containing A . And, there is a continuum K and a neighborhood V of a point x of A such that V ⊂ IntK ⊂ K ⊂ U . And Let L ₁ , L ₂ ∈< U ₁ , ⋯ , U_n , V > ∩ C ( X ). Then L ₁ ∩ K ≠ ø and L ₂ ∩ K ≠ ø. It follows that L ₁ ∪ L ₂ ∪ K ∈< U ₁ , ⋯ , U_n , V > ∩ C ( X ). Hence there is order arcs L ₁ and L ₂ in < U ₁ , ⋯ , U_n , K > ∩ C ( X ) from L ₁ to L ₁ ∪ L ₂ ∪ K and from L ₂ to L ₁ ∪ L ₂ ∪ K . It follows that there is an arc in L ₁ ∪ L ₂ from L ₁ to L ₂ , and it is clear that L ₁ ∪ L ₂ ⊂< U ₁ , ⋯ , U_n , V > ∩ C ( X ). Therefore C ( X ) is locally arcwise connected at A . ☐ Theorem 2.2. Let A ∈ C ( X ). If IntA ≠ ø, then for each open set U containing A there is a continuum K and a neighborhood V of a point of A such that V ⊂ IntK ⊂ K ⊂ U . Proof. Let U be an open set containing A . Let x ∈ IntA . Then there is an open set V such that x ∈ V ⊂ IntA , and hence x ∈ IntA ⊂ A ⊂ U . In this case A is a continuum which satisfies the condition of the continuum K in this theorem. ☐ We get the below Corollary from Theorem 2.1 and Theorem 2.2. Corollary 2.3 ([Theorem 3 of [4] ]). Let A ∈ C ( X ). If IntA ≠ ø, then C ( X ) is locally arcwise connected at A . Proof. Let A ∈ C ( X ) and let < U ₁ , ⋯ , U_n < ∩ C ( X ) be a basic open set containing A . Let x ∈ IntA and let V be an open set such that x ∈ V ⊂ IntA and such that Then A ∈< U ₁ , ⋯ , U_n , V >⊂< U ₁ , ⋯ , U_n >. Let L ₁ , L ₂ ∈< U ₁ , ⋯ , U_n , V > ∩ C ( X ). Then L ₁ ∩ V ≠ ø and L ₂ ∩ V ≠ ø, so L ₁ ∩ A ≠ ø and L ₂ ∩ A ≠ ø. It follows that L ₁ ∪ L ₂ ∪ A ∈< U ₁ , ⋯ , U_n , V > ∩ C ( X ). Hence there is order arcs L ₁ and L ₂ in < U ₁ , ⋯ , U_n , V > ∩ C ( X ) from L ₁ to L ₁ ∪ L ₂ ∪ A and from L ₂ to L ₁ ∪ L ₂ ∪ A . It follows that there is an arc in L ₁ ∪ L ₂ from L ₁ to L ₂ , and it is clear that L ₁ ∪ L ₂ ⊂< U ₁ , ⋯ , U_n , V > ∩ C ( X ). ☐ Theorem 2.4. Let X be a Hausdorff continuum, and let A ∈ C ( X ). If X is connected im kleinen at A, then A is admissible. Proof . Let x ∈ A ∈ C ( X ) and X is connected im kleinen at A . Let < U ₁ , ⋯ , U_n > ∩ C ( X ) be a basic open set containing A , and let . Then A ⊂ U and there is a continuum K such that A ⊂ IntK ⊂ K ⊂ U . Set V_x = IntK . Then for every y ∈ V_x , y is an element of K . And since A ⊂ K and K ⊂ U , K ∈< U ₁ , ⋯ , U_n > ∩ C ( X ). Thus A is admissible. ☐ Theorem 2.5. Let X be a Hausdorff continuum, and let A ∈ C ( X ). If A is admissible, then for any open subset U of C ( X ) containing A, there is an open subset V of X such that . Proof. Let U be an open set containing A in C ( X ), and let x ∈ A . Then by the definition of admissibility there is an open set V_x containing x in X such that for every y ∈ V_x there is a continuum B in C ( X ) such that y ∈ B ∈ U . Set . Then . Theorem 2.6. Let X be a Hausdorff continuum, and let A ∈ C ( X ). If for any open subset U of C ( X ) containing A, there is a subcontinuum K of X such that A ∈ IntK ⊂ K ⊂ U and there is an open subset V of X such that then A is admissible. Proof. Let U =< U ₁ , ⋯ , U_n > ∩ C ( X ) be a basic open subset of C ( X ) containing A , let K a continuum in C ( X ) contains A in its interior, let V an open subset of X such that . Then for any element y of V , is a continuum in U containing y .