Definition
A topological space is said to be σ-orthocompact iff every open cover has a σ-interior preserving (i.e. countable union of interior preserving? families) open refinement.
Reference
Jianjun Wang and Peiyong Zhu, σ-orthocompactness in product spaces, Internat. J. Math. Comp. Sci. 6:1 (2010)