prove the following closed set S setS union boundary SSolut

prove the following: closed set S = setS union boundary S

Solution

In a topological space, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if and only if it contains all of its limit points.

The closed interval [a,b] of real numbers is closed.

    In general topology, you cannot say that the boundary of the union is contained in the union of the boundaries. For instance, enumerate the rationals in (0,1) by R={rk}. Then the boundary of the union is [0,1], but the union of the boundaries is R itself. it may be the case that the boundary of the union is contained in the boundary of the unions.

     so that ,

            closed set S = setS union boundary S