$M$-fuzzifying convexity-preserving mappings and $M$-fuzzifying closure-preserving mappings

In this paper, $M$-fuzzifying convexity-preserving mappings between $M$-fuzzifying convergence spaces, and $M$-fuzzifying closure-preserving mappings between $M$-fuzzifying preconvex closure spaces are proposed. The relationships of $M$-fuzzifying convexity-preserving mappings with $M$-CP mappings, $M$-fuzzifying preconvex closure operators, and separation properties in $M$-fuzzifying convergence spaces are discussed. Moreover, it is proved that $S_0$, $S_1$ and $S_2$ separation properties are preserved by homeomorphisms in $M$-fuzzifying convergence spaces.