Proposition 2.33.g

Proposition 2.33.g: $$Bd(A) = Bd(A^c)$$

Olomana
We know that $$(A^c)^c = A$$. Substitute $$A^c$$ for $$A$$ in this definition:

Call $$x$$ an boundary point of $$A$$ if every open set $$U$$ containing $$x$$ contains a point in $$A$$ and a point in $$A^c$$.

Simplify and rearrange:

Call $$x$$ an boundary point of $$A^c$$ if every open set $$U$$ containing $$x$$ contains a point in $$A$$ and a point in $$A^c$$.

The definitions are the same, so $$Bd(A)= Bd(A^c)$$.

Olomana 02:07, March 21, 2011 (UTC)