We define the nearly submetacompact space and give a characterization of it.

This paper proves the following results:Let X=lim→{Xα,παβ,Λ},λ=|Λ| and each projection πα is an open and onto mapping for each α∈Λ,if X is λ-paracompact and each Xα is nearly submetexpandable,then X is nearly submetexpandable.

Nearly submetacompact spaces and nearly strongly submetacompact spaces;

A new characterization of the almost homogeneous functions is obtained which can be considered as a natural extension of the Euler equatio n characterizing the homogeneous functions.