Proposition 2.33.f

Proposition 2.33.f: $$Bd(A) = Cl(A) \setminus Int(A)$$.

Ian_mi
By proposition 2.33b, $$\text{Cl}(A) = \text{Int}(A) \cup \text{Bd}(A)$$ so $$\text{Cl}(A) \setminus \text{Int}(A) = \text{Bd}(A) \setminus \text{Int}(A)$$. Therefore it suffices to show that $$\text{Bd}(A) \setminus \text{Int}(A) = \text{Bd}(A)$$. Let $$x$$ a boundary point of $$A$$ and let $$G$$ be an open set containing $$x$$. Then $$G$$ contains a point of $$A^c$$ so $$G \not \subseteq A$$. Therefore $$x \notin \text{Int}(A)$$ so $$\text{Bd}(A) \setminus \text{Int}(A) = \text{Bd}(A)$$. This proves the proposition.

Ian mi 03:30, March 24, 2011 (UTC)Ian_mi