Separation Axioms and Lattice Equivalence

Abstract

>This paper deals with the relation between lattice-equivalence and some separation axioms. We are concerned with two questions: The first one is to characterize topological spaces $X$ such that $X$ and $\mathbf{F}(X)$ are lattice equivalent for some covariant functors $\mathbf{F}$ from $\mathbf{TOP}$ to itself. In the second question, it is proved that $T_{(0,2)}, T_{(S,D)}, T_{(S,1)}$ and $T_{(0,3\frac{1}{2})}$ are lattice-invariant properties but $S$, $T_{(0,1)}$, $T_{(0,S)}$, $T_{(1,2)}$, $T_{(1,S)}$, $T_{(1,3\frac{1}{2})}$, and $T_{(0,D)}$ are not.