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LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Boundary points

Let A be a set. Denote by dA the set of boundary points of A. Denote by intA the interior of A. Denote A' the closure of A.

It was asked of me to prove the following:

(i) dA' is a subset of dA
(ii) d(intA) is a subset of dA

and I have already done so. However, now I'm asked to find a example of a set A such that dA, dA', and d(intA) are all different. This set can be as weird as you like and you can consider any topology on any space.

I been trying to find such a set for some time but haven't been able. Such a set A cannot be open because then A = intA and obviously their boundary points will be the same. Similarly if A is closed A = A'

Last edited by LuisRodg (2011-02-12 12:20:11)

Alg Num Theory

Member

Registered: 2017-11-24

Posts: 693

Website

Re: Boundary points

Try

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Then

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and

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so

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Me, or the ugly man, whatever (3,3,6)