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Abstract

For a Tikhonov space X we denote by Pc(X) the semilattice of all continuous pseudometrics on X. It is proved that compact Hausdorff spaces X and Y are homeomorphic if and only if there is a positive-homogeneous (or an additive) semi-lattice isomorphism T:Pc(X) → Pc(Y). A topology on Pc(X) is called admissible if it is intermediate between the compact-open and pointwise topologies on Pc(X). Another result states that Tikhonov spaces X and Y are homeomorphic if and only if there exists a positive-homogeneous (or an additive) semi-lattice homeomorphism $T:(Pc(X),τ_X) → (Pc(Y),τ_Y)$, where $τ_X,τ_Y$ are admissible topologies on Pc(X) and Pc(Y).