Previous |  Up |  Next


linear functorial operator extending (pseudo)metrics; the functor of $G$-symmetric power
For a functor $F\supset Id$ on the category of metrizable compacta, we introduce a conception of a linear functorial operator $T=\{T_X:Pc(X)\to Pc(FX)\}$ extending (for each $X$) pseudometrics from $X$ onto $FX\supset X$ (briefly LFOEP for $F$). The main result states that the functor $SP^n_G$ of $G$-symmetric power admits a LFOEP if and only if the action of $G$ on $\{1,\dots,n\}$ has a one-point orbit. Since both the hyperspace functor $\exp$ and the probability measure functor $P$ contain $SP^2$ as a subfunctor, this implies that both $\exp$ and $P$ do not admit LFOEP.
Banakh T.: AE(0)-spaces and regular operators extending (averaging) pseudometrics. Bull. Polon. Acad. Sci. Ser. Sci. Math. (1994), 42 197-206. MR 1811849 | Zbl 0827.54010
Bessaga C., Pełczyński A.: On the spaces of measurable functions. Studia Math. 44 (1972), 597-615. MR 0368068
Fedorchuk V.V.: On some geometric properties of covariant functors (in Russian). Uspekhi Mat. Nauk 39 (1984), 169-208. MR 0764014
Fedorchuk V.V.: Triples of infinite iterates of metrizable functors (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. (1990), 54 396-418.
Fedorchuk V.V., Filippov V.V.: General Topology. Principal Constructions (in Russian). Moscow Univ. Press Moscow (1988).
Pikhurko O.: Extending metrics in compact pairs. Mat. Studiï 3 (1994), 103-106. MR 1692801 | Zbl 0927.54029
Zarichnyi M.: Regular linear operators extending metrics: a short proof. Bull. Polish. Acad. Sci. 44 (1996), 267-269. MR 1419399 | Zbl 0866.54017
Partner of
EuDML logo