Question

Let E^o denote the set of all interior points of a set E.

Prove: E is open if and only if E^o= E

Answer 1

Open set: Let A set said to be Open if all x belong to A , there exist a Delta neighborhood such that, that neighborhood contained in A.

Let Eo denotes the seb of alt iuterioy PointT of sel E Claim Eis open iff Eo= E then a e E suppose E ir open If for these exist d70 any def of open set). a is iuterioy point of E a e E o Е СЕо But iuterioY point set of are the subseb ef E any Neighbous hood of poiuts in E since in Eo wil be from Convers aly f Eo = E clai m i E is open It ir gufficient to Show suppose b E Eo There existr d o 5.t Ncb,d Since any nelghbourhood of poiutr ar VCo,d is Open CENC6,d, there ex ist Ero СВЕ. op en set So for any NCCECNCo,d) K(0def of open set NC,d) CE ul NCC,E) CE poind in N C6, i in Thio means Cing evy aiuteriorof E NC6,J)CEO E i open