If each is a Hausdorff space, then is a Hausdorff space in both the box and product top.
How do I go about proving this?
I know since is Hausdorff, if for all where there exists neighborhoods of that are disjoint.
If and are distinct points in the product space, then they must differ in at least one coordinate. So there exists say, such that . Use the Hausdorff property in the -coordinate space to separate and by disjoint neighbourhoods. Then use those neighbourhoods to construct neighbourhoods in the product space that separate and . The same construction will work for both the product and the box topologies.