We prove that any infinite-dimensional non-archimedean Fréchet space $E$ is homeomorphic to $D^{\mathbb{N}}$ where $D$ is a discrete space with $\mathop {\mathrm card}(D)=\mathop {\mathrm dens}(E)$. It follows that infinite-dimensional non-archimedean Fréchet spaces $E$ and $F$ are homeomorphic if and only if $\mathop {\mathrm dens}(E)= \mathop {\mathrm dens}(F)$. In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field $\mathbb{K}$ is homeomorphic to the non-archimedean Fréchet space $\mathbb{K}^{\mathbb{N}}$.
Photosynthetic utilization of radiant energy was studied by chlorophyll (Chl) fluorescence and maximum photosynthetic O2 evolution (Pmax) in temperate lettuce (Lactuca sativa L.) grown under natural tropical fluctuating ambient temperatures but with their roots exposed to two different root-zone temperatures (RZTs): a constant 20 °C-RZT (RZT20) and a fluctuating ambient RZT (RZTa) from 23 to 40 °C. On a sunny day, irrespective of RZT, ΔF/Fm' [ratio of the variable to maximal fluorescence under irradiation (the maximal photosystem 2 quantum yield with "actinic light")] decreased and non-photochemical quenching (NPQ) increased parallel to the increase of photosynthetic photon flux density (PPFD). However, RZTa plants showed lower ΔF/Fm' and higher NPQ than RZT20 plants. The electron transport rate (ETR) was much higher in RZT20 plants than in RZTa plants especially during moderately sunny days. There were no significant diurnal changes in Pmax although these values of RZT20 plants were much higher than those of RZTa plants. On cloudy days, no significant diurnal changes in ΔF/Fm' and NPQ occurred, but ΔF/Fm' was higher and NPQ was lower in RZT20 plants than in RZTa plants. Diurnal changes in ETR were also observed in all plants while Pmax values throughout the whole cloudy days in both RZT20 and RZTa plants were constant. Again, RZT20 plants had much higher values of Pmax than RZTa plants. During RZT transfer period, all Chl fluorescence parameters measured at midday fluctuated with PPFD. Impact of RZT on these parameters was observed 2-3 d after RZT transfer. ETR and Pmax measured with saturating PPFD in the laboratory did not vary with the fluctuating PPFD in the greenhouse but the effects of RZT on these two parameters were observed 3-4 d after RZT transfer. Thus RZT affects photosynthetic utilization of photon energy in temperate lettuce grown under natural tropical condition. and J. He, S. K. Lee.