Pointwise topological convergence and topological graph convergence of set-valued maps

Let X,Y be topological spaces and {Fn : n ? ?} be a sequence of set-valued maps from X to Y with the pointwise topological limit G and with the topological graph limit F. We give an answer to the question from ([19]): which conditions on X,Y and/or {F,G,Fn : n ? ?} are needed to F = G.

Variation of aromaticity by twisting or expanding the ring content

Generalization of the Hückel rule predicts that the (anti)aromaticity of a neutral ring is qualitatively reverted upon a single twist of the π-orbital array (Möbius interconversion), and is preserved upon expansion of all the bonds by single C2 units (ring carbo-merization). These opposite effects are addressed from quantitative theoretical and experimental standpoints, respectively. (i) According to most resonance energy (RE) schemes, the RE value of a Möbius ring is not the opposite of that of the Hückel version. This also applies to the Aihara’s and Trinajstic’s topological resonance energy (TRE), where a non-aromatic reference in the topological limit is defined as being “as identical as possible” to the parent ring but just “acyclic”. In spite of its conceptual merits, the computing complexity and fictitious character of the TRE acyclic reference resulted in a disuse of TRE as a current energetic aromaticity index. Both the calculation and interpretation of TRE have been revisited in light of a cross-reference between the Hückel and Möbius rings within the Hückel molecular orbital (HMO) framework. Whereas the topological influence of triple bonds is currently neglected in the first-level HMO treatment of π-conjugated systems, a graph-theoretical analysis allows one to differentiate the TRE value of a [3n]annulene from those of the corresponding carbo-[n]annulene. The C18 ring of carbo-benzene is thus predicted to be slightly more topologically aromatic than that of [18]annulene. (ii) Recent experimental and density functional theory (DFT) theoretical studies of quadrupolar carbo-benzene derivatives are presented. The results show that the “flexible aromaticity” of the p-C18Ph4 bridge between donor anisyl substituents plays a crucial role in determining the intriguing chemical/spectroscopical/optical properties of these carbo-chromophores.

DUALITY AND THE EQUIVALENCE PRINCIPLE OF QUANTUM MECHANICS

Following a suggestion by Vafa, we present a quantum-mechanical model for S duality symmetries observed in the quantum theories of fields, strings and branes. Our formalism may be understood as the topological limit of Berezin's metric quantization of the upper half-plane H, in that the metric dependence of Berezin's method has been removed. Being metric-free, our prescription makes no use of global quantum numbers. Quantum numbers arise only locally, after the choice of a local vacuum to expand around. Our approach may be regarded as a manifestly nonperturbative formulation of quantum mechanics, in that we take no classical phase space and no Poisson brackets as a starting point. Position and momentum operators satisfying the Heisenberg algebra are defined and their spectra are analysed. We provide an explicit construction of the Hilbert space of states. The latter carries no representation of SL (2,R), due to the lifting of the metric dependence. Instead, the reparametrization invariance of H under SL (2,R) induces a natural SL (2,R) action on the quantum-mechanical operators that implements S duality. We also link our approach with the equivalence principle of quantum mechanics recently formulated by Faraggi–Matone.

Notes on general topology

It has been known for some time that the product of a non-metrizable Hausdorff space and any (non-trivial) Hausdorff space cannot be the continuous image of an ordered continuum. (For a survey of this and related problems, see Mardešić and Papić ((1)).) Further, it has been shown by Treybig ((2)) (and independently by the present author) that the product of two Hausdorff spaces cannot even be the continuous image of an ordered compactum unless both the spaces are metrizable or one is finite. It is therefore of some interest to give a simple example of a space X which is the continuous image of an ordered compactum K and contains the product of a non-metrizable space and an infinite discrete space, imbedded in such a way as to form a sequence of homeomorphic subsets with a connected (non-trivial) topological limit.