# An example of an Alexandrov topology on an infinite set that is not connected.

In topology, an Alexandrov topology is a topology in which the intersection of ANY family of open sets is open. I have determined that the connected components of an Alexandrov topology are necessarily both open and closed. Can someone give an example of an Alexandrov topology on an infinite set that is not connected?

• Discrete topology ... – Hagen von Eitzen Apr 15 '17 at 14:09
• The discrete space on an infinite set is Alexandrov. So $\mathbb{N}$ together with its powerset $P(\mathbb{N})$, is an example of such a space – Mike Apr 15 '17 at 14:10