This paper first prove an inverse limit theorem of strongly submetacompact spaces; By using this, two Tychonoff product theorems of infinit factor spaces are obtained; Finally, it is proved that the equalentness on strongly submetacompact spaces and submetacompact spaces is proved.

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

オetacompact space and submetacompact space are especially studied in the classes of orthocompact space and suborthocompact space,two representing theorems are gained;a theorem of Junnila is generalized,a characterization of submetacompact space is gained.

The author mainly obtained the following two theorems: (1)Let X=σ{x α:α∈A},if every finite subproduct of X is strong submetacompact, then X is strong submetacompact; (2) Let X=σ{x α:α∈A}, if every finite subproduct of X is heredifarily submetacompact and X normal, then X is hereditarily submetacompact.

In this paper,we show the result:A space X is a hereditarily submetacompact space if and only if every scattered partition of X has a θ-sequence of open expansions.