Real Analysis: Interior, Closure and Boundary

[michael.wes]

Homework Statement

Let $$W\subset S \subset \mathbb{R}^n.$$ Show that the following are equivalent: (i) $$W$$ is relatively closed in $$S$$, (ii) $$W = \bar{W}\cap S$$ and (iii) $$(\partial W)\cap S \subset W$$.

Homework Equations

The only thing we have to work with is the definitions of open and closed sets, relatively open and relatively closed sets, the result that the complement of an open set is closed, the definition of boundary, interior and closure.

The Attempt at a Solution

I have proved that (ii) implies (i). I need help with (i) implies (iii) and (iii) implies (ii).

For (i) implies (iii), there exists a closed set $$C$$ such that $$W=C\cap S$$. Now try to derive that $$\partial W \cap S \subset W$$. Write $$\partial(C\cap S)\cap S$$, but after this all I am able to do is draw a diagram and get stuck.

For (iii) implies (ii), we have that $$\partial W \cap S \subset W$$ and we need to derive $$W = (W\cup\partial W) \cap S$$, but I can't get started.

Please help. Thank you!

 

[ystael]

Try the direction (i) => (ii) => (iii) => (i) instead. You will need the relation $$\overline{W} = W^\circ \cup \partial W$$.