On The Relationships Between Types of $L$-convergence Spaces

This paper focuses on the relationships between stratified $L$-conver-gence spaces, stratified strong $L$-convergence spaces and stratifiedlevelwise  $L$-convergence spaces. It has been known that: (1) astratified $L$-convergence space is precisely a left-continuousstratified levelwise $L$-convergence space; and (2) a  stratifiedstrong $L$-convergence space is naturally a stratified $L$-convergence space, but the converse is not true generally.In this paper, a strong left-continuity condition for stratified levelwise $L$-convergence space is given. It is proved that  a stratified strong $L$-convergence space  is precisely a strongly left-continuous  stratifiedlevelwise $L$-convergence space. Then a sufficient and necessary condition for a stratified $L$-convergence space to be a  stratified strong $L$-convergence space is presented.