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A partially ordered set, is -chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an o> -chain complete partially ordered set has a least element, denoted by 0.A function / mapping a partially ordered set P into a partially ordered set Q is chain continuous if for any nonempty chain C in P, which has a supremum, /(sup P C) = sup Q /(C). It is o>-chain continuous if for any nonempty countable chain C in P, which has a supremum, /(sup P C) = sup Q /(C). A partially ordered set P has the least fixed point property if every order-preserving function from P to itself has a least fixed point. It has the fixed point property if every order-preserving function from P to itself has a fixed point. It has the least fixed point property for o>-chain continuous functions if every -chain completeness follows from the least fixed point property for a) -chain continuous functions. Kolodner has shown in (5) that the converse is true. The following example shows that the answer to Plotkin's question is no. Example 1. Let P l be the partially ordered set defined by the diagram on the next page. Let C = {c n |neN}, X = { x J n e N } and Y = {y n |neN}. Then x = s u p X and C has no supremum in P. Proposition 1. Every

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Abstract. The techniques developed by Wage and Norden are used to show that the Pixley-Roy hyperspaces of any two «-graphs are homeomorphic. The Pixley-Roy hyperspaces of several subsets of R" are also shown to be homeomorphic.

Increasing interest in the thermodynamics of small and/or isolated systems, in combination with recent observations of negative temperatures of atoms in ultracold optical lattices, has stimulated the need for estimating the conventional, canonical temperature T of systems in equilibrium with heat baths using eigenstate-specific temperatures (ESTs). Four distinct ESTs-continuous canonical, discrete canonical, continuous microcanonical, and discrete microcanonical-are accordingly derived for two-level paramagnetic spin lattices (PSLs) in external magnetic fields. At large N, the four ESTs are intensive, equal to T, and obey all four laws of thermodynamics. In contrast, for N < 1000, the ESTs of most PSL eigenstates are non-intensive, differ from T, and violate each of the thermodynamic laws. Hence, in spite of their similarities to T at large N, the ESTs are not true thermodynamic temperatures. Even so, each of the ESTs manifests a unique functional dependence on energy which clearly specifies the magnitude and direction of their deviation from T; the ESTs are thus good temperature estimators for small PSLs. The thermodynamic uncertainty relation is obeyed only by the ESTs of small canonical PSLs; it is violated by large canonical PSLs and by microcanonical PSLs of any size. The ESTs of population-inverted eigenstates are negative (positive) when calculated using Boltzmann (Gibbs) entropies; the thermodynamic implications of these entropically induced differences in sign are discussed in light of adiabatic invariance of the entropies. Potential applications of the four ESTs to nanothermometers and to systems with long-range interactions are discussed.