Theorem 1 proves (among the others) that for a locally compact topological group X the following assertions are equivalent: (i) X is metrizable and σ-compact. (ii) Cp(X) is analytic. (iii) Cp(X) is K-analytic. (iv) Cp(X) is Lindelöf. (v) Cc(X) is a separable metrizable and complete locally convex space. (vi) Cc(X) is compactly dominated by irrationals. This result supplements earlier results of Corson, Christensen and Calbrix and provides several applications, for example, it easily applies to show that: (1) For a compact topological group X the Eberlein, Talagrand, Gul'ko and Corson compactness are equivalent and any compact group of this type is metrizable. (2) For a locally compact topological group X the space Cp(X) is Lindelöf iff Cc(X) is weakly Lindelöf. The proofs heavily depend on the following result of independent interest: A locally compact topological group X is metrizable iff every compact subgroup of X has countable tightness (Theorem 2). More applications of Theorem 1 and Theorem 2 are provided.
Raman spectroscopic data were obtained for (Mg,Fe)2SiO4 samples during compression to 57 GPa. Single crystals of San Carlos olivine compressed hydrostatically above 41 GPa showed appearance of a new “defect” peak in the 820–840 cm–1 region associated with SiOSi linkages appearing between adjacent tetrahedra to result in five- or sixfold-coordinated silicate species. Appearance of this feature is accompanied by a broad amorphous background. The changes occur at lower pressure than metastable crystalline transitions of end-member Mg2SiO4 forsterite (Fo-I) into Fo-II and Fo-III phases described recently. We complemented our experimental study using density functional theory (DFT) calculations and anisotropic ion molecular dynamics (AIMD) simulations to study the Raman spectra and vibrational density of states (VDOS) of metastably compressed Mg2SiO4 olivine, Fo-II and Fo-III, and quenched melts at high and low pressures. By 54 GPa all sharp crystalline peaks disappeared from observed Raman spectra indicating complete pressure-induced amorphization (PIA). The amorphous (Mg,Fe)2SiO4 spectrum contains Si-O stretching bands at lower wavenumber than expected for indicating high coordination of the silicate units. The amorphous spectrum persisted on decompression to ambient conditions but with evidence for reappearance of tetrahedrally coordinated units. Non-hydrostatic compression of polycrystalline olivine samples showed similar appearance of the defect feature and broad amorphous features between 43–44 GPa. Both increased in intensity as the sample was left at pressure overnight but they disappeared during decompression below 17 GPa with recovery of the starting olivine Raman signature. A hydrated San Carlos olivine sample containing 75–150 ppm OH was also studied. Significant broadening of the stretching peaks was observed above 43 GPa but without immediate appearance of the defect or broad amorphous features. However, both of these characteristics emerged after leaving the sample at 47 GPa overnight followed by complete amorphization that occurred upon subsequent pressurization to 54 GPa. During decompression the high-density amorphous spectrum was retained to 3 GPa but on final pressure release a spectrum similar to thermally quenched low-pressure olivine glass containing isolated groups was obtained. Leaving this sample overnight resulted in recrystallization of olivine. Our experimental data provide new insights into the metastable structural transformations and relaxation behavior of olivine samples including material recovered from meteorites and laboratory shock experiments.