A Proposition
Posted: Sun Jun 19, 2016 11:29 am
Let \(\displaystyle{V}\) be an algebraic subset of \(\displaystyle{\mathbb{A}^n}\).
1. The points of \(\displaystyle{V}\) are closed for the \(\displaystyle{\rm{Zariski}}\) - topology.
2. Every ascending chain \(\displaystyle{\left(U_n\right)_{n\in\mathbb{N}}}\) of open subsets
of \(\displaystyle{V}\) eventually becomes constant. Equivalently, every descending chain of closed
subsets of \(\displaystyle{V}\) eventually becomes constant.
3. Every open covering of \(\displaystyle{V}\) has a finite subcovering.
1. The points of \(\displaystyle{V}\) are closed for the \(\displaystyle{\rm{Zariski}}\) - topology.
2. Every ascending chain \(\displaystyle{\left(U_n\right)_{n\in\mathbb{N}}}\) of open subsets
of \(\displaystyle{V}\) eventually becomes constant. Equivalently, every descending chain of closed
subsets of \(\displaystyle{V}\) eventually becomes constant.
3. Every open covering of \(\displaystyle{V}\) has a finite subcovering.