About The Prime Ends And Some Properties Of The Prime Ends

The theory of prime ends was developed by Caratheodory. This paper shows that the definition and examples of prime end and gives the connectivity and properties of impression.THEOREM 1 let U ? C) be a simply connected domain such that K = C)\U contains more than one point and let P be a prime end of U . Then the impression I ( p) contains at most two points at which K is locally connected.THEOREM 2 The set K is locally connected if and only if every prime end impression is trivial.THEOREM 3 Let Q1 , Q2,L QnLbe a sequence of chains such that(a) for all n, Q n+1is a looped refinement of Q n(b) for all n, the radius of each link of Q n isTHEOREM 4 Let f be a conformal map of D onto G and let the prime end P be symmetric, Ifξ≠ξ′,and if E is a continuum withAt the same time, it shows properties of that impression of the prime end . This is useful for study of the boundary.