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Recall that a collection 9 of subsets of X is point-countable if every x G X is in at most countably many ?Gf. Such collections have been studied from several points of view. First, in characterizing various kinds of j-images of metric spaces, second, to construct conditions which imply that compact spaces and some of their generalizations are metrizable, and finally, in the context of meta-Lindelόf spaces. This paper will make some contributions to all of these areas. Introduction. A space1 X is determined by a cover £P, or 9 determines X, 2 if U C X is open (closed) in X if and only if U Π P is relatively open (relatively closed) in P for every P E <•?. (For example, X is determined by <$ if <$ is locally finite or if {P°: P E <3>} covers X.) We will explore this notion in the context of point-countable covers.Within the class of λ -spaces, spaces determined by point-countable covers turn out to be related to spaces having a point-countable Λ -network. Recall that a cover ζP of X is a k-network for X if, whenever K C U with K compact and U open in X 9 then K C U φ C U for some finitê C^. Such collections have played a role in S 0 -spaces (i.e., regular spaces with a countable λ -network [MJ) and K-spaces (i.e., regular spaces with a σ-locally finite λ -network [0]).

We present the measurement and analysis of five new (D 2 O) 5 bands via vibration-rotation-tunneling (VRT) spectroscopy as well as a preliminary description of a second (H 2 O) 5 band. The vibrationally averaged rotational constants of all five fitted bands agree well with those from the two previously observed (D 2 O) 5 bands and confirm that the pentamer averages to a symmetric, quasi-planar structure on the time scale of our experiment. While the spectrum of the first two bands, located at 50.7 cm -1 (1.52 THz) and 27.3 cm -1 (0.82 THz) are indicative of unperturbed oblate rotors, the three remaining (D 2 O) 5 bands centered at 47.7 cm -1 (1.43 THz), 45.4 cm -1 (1.36 THz), and 45.0 cm -1 (1.35 THz) are severely perturbed by first-order Coriolis coupling. This represents the first observation of this perturbation in the perdeuterated water pentamer, as well as the first observation of transitions between degenerate states of the torsional-puckering manifold. Unlike transitions from the (H 2 O) 5 band observed by Brown et al. at 89.0 cm -1 and the 103.8 cm -1 band that we report here, none of the individual rovibrational transitions of any of the five (D 2 O) 5 bands demonstrate spectral splittings due to bifurcation tunneling. We conclude, through careful analysis of these water pentamer bands, that at least three torsional-puckering manifolds have been probed and that the lowest-energy manifold is highly compacted. A plausible water pentamer torsional-puckering correlation diagram is proposed, though additional experimental data are required to unambiguously establish the energies of the torsional-puckering levels.

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