Orthocompact space

In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open cover has an interior preserving open refinement. 
That is, given an open cover of the topological space, there is a refinement which is also an open cover, with the further property that at any point, the intersection of all open sets in the refinement containing that point, is also open.
If the number of open sets containing the point is finite, then their intersection is clearly open. That is, every point finite open cover is interior preserving. 
Hence, we have the following: every metacompact space, and in particular, every paracompact space, is orthocompact.
Useful theorems: