Proposition 2.33.d

Proposition 2.33.d: $$Int(A) = (Cl(A^c))^c$$.

Ian_mi
By proposition 233b, 2.33c, and 2.33g, $$\text{Int}(A)^c = (A \setminus \text{Bd}(A))^c = A^c \cup \text{Bd}(A) = A^c \cup \text{Bd}(A^c) = (A^c \setminus \text{Bd}(A^c)) \cup \text{Bd}(A^c) = \text{Int}(A^c) \cup \text{Bd}(A^c) = \text{Cl}(A^c)$$. Therefore $$\text{Int}(A) = \text{Cl}(A^c)^c$$.

Ian mi 03:20, March 24, 2011 (UTC)Ian_mi