Closed and compact
Posted: Wed Jun 08, 2016 4:31 pm
Let \(\displaystyle{X}\) be a topological vector space, \(\displaystyle{K\subseteq X}\) a compact
subset of \(\displaystyle{X}\) and \(\displaystyle{C\subseteq X}\) a closed subset of \(\displaystyle{X}\)
such that \(\displaystyle{K\cap C=\varnothing}\).
Then, there exists an open region \(\displaystyle{V}\) of \(\displaystyle{0\in X}\) such that
\(\displaystyle{(K+V)\cap (C+V)=\varnothing}\)
subset of \(\displaystyle{X}\) and \(\displaystyle{C\subseteq X}\) a closed subset of \(\displaystyle{X}\)
such that \(\displaystyle{K\cap C=\varnothing}\).
Then, there exists an open region \(\displaystyle{V}\) of \(\displaystyle{0\in X}\) such that
\(\displaystyle{(K+V)\cap (C+V)=\varnothing}\)