Proposition 2.36

Proposition 2.36: $$Ext(A) = Int(A^c)$$.

Olomana
Note that $$U \subseteq A$$ is logically equivalent to $$U \cap A^c = \emptyset$$. This allows us to rewrite the definition of an interior point as:

Call $$x$$ an interior point of $$A$$ if there exists an open set $$U$$ containing $$x$$ such that $$U \cap A^c = \emptyset$$.

Now the symmetry between interior points and exterior points is clear:

Call $$x$$ an exterior point of $$A$$ if there exists an open set $$U$$ containing $$x$$ such that $$U \cap A = \emptyset$$.

Since $$(A^c)^c = A$$, we can substitute $$A^c$$ for $$A$$ in the definition of an interior point, and get $$Ext(A) = Int(A^c)$$.

Olomana 05:50, March 19, 2011 (UTC)

'''That does it. --Steven.clontz 21:10, March 19, 2011 (UTC)'''