Corollary 2.29

Corollary 2.29: A set $$U$$ is OPEN (the main text has a typo) in $$X$$ if and only if $$U = Int(U)$$.

Olomana
First, if $$U$$ is open, then $$U$$ itself is an open subset of $$U$$. By 2.27, $$Int(U)$$ is the union of all open subsets of $$U$$, including $$U$$ itself, and this union is $$U$$.

Conversely, if $$U$$ is the union of all open subsets of $$U$$, then $$U$$ itself is open (topologies are closed under arbitrary union).

Olomana 06:56, March 20, 2011 (UTC)