Sur la caractérisation topologique des compacts à l'aide des demi-treillis des pseudométriques continues

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 ${\displaystyle T:(Pc\left(X\right),{\tau }_X)\to (Pc\left(Y\right),{\tau }_Y)}$, where ${\displaystyle {\tau }_X,{\tau }_Y}$ are admissible topologies on Pc(X) and Pc(Y).