1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Real Analysis: Interior, Closure and Boundary

  1. Jan 21, 2010 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Let [tex] W\subset S \subset \mathbb{R}^n.[/tex] Show that the following are equivalent: (i) [tex]W[/tex] is relatively closed in [tex]S[/tex], (ii) [tex]W = \bar{W}\cap S[/tex] and (iii) [tex](\partial W)\cap S \subset W[/tex].

    2. Relevant 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.

    3. 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 [tex]C[/tex] such that [tex]W=C\cap S[/tex]. Now try to derive that [tex]\partial W \cap S \subset W[/tex]. Write [tex]\partial(C\cap S)\cap S[/tex], but after this all I am able to do is draw a diagram and get stuck.

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

    Please help. Thank you!
  2. jcsd
  3. Jan 21, 2010 #2
    Try the direction (i) => (ii) => (iii) => (i) instead. You will need the relation [tex]\overline{W} = W^\circ \cup \partial W[/tex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Similar Threads for Real Analysis Interior Date
If A is dense in [0,1] and f(x) = 0, x in A, prove ∫fdx = 0. Friday at 11:51 PM
Real Analysis Definition and Explanation Mar 27, 2018
Real Analysis Proof Mar 11, 2018
Rudin POMA: chapter 4 problem 14 Feb 28, 2018
Interior Point of 1/n Nov 11, 2015