On classes of T0 spaces admitting completions

[EN] For a given class X of T0 spaces the existence of a subclass C, having the same properties that the class of complete metric spaces has in the class of all metric spaces and non-expansive maps, is investigated. A positive example is the class of all T0 spaces, with C the class of sober T0 spaces, and a negative example is the class of Tychonoff spaces. We prove that X has the previous property (i.e., admits completions) whenever it is the class of T0 spaces of an hereditary coreflective subcategory of a suitable supercategory of the category Top of topological spaces. Two classes of examples are provided.Giuli, E. (2003). On classes of T0 spaces admitting completions. Applied General Topology. 4(1):143-155. doi:10.4995/agt.2003.2016.SWORD1431554

Functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics

[EN] This paper studies various functors between (lattice-valued) topology and (lattice-valued) bitopology, including the expected “doubling” functor Ed : L-Top → L-BiTop and the “cross” functor E× : L-BiTop → L2-Top introduced in this paper, both of which are extremely well-behaved strict, concrete, full embeddings. Given the greater simplicity of lattice-valued topology vis-a-vis lattice-valued bitopology and the fact that the class of L2-Top’s is strictly smaller than the class of L-Top’s encompassing fixed-basis topology, the class of E×’s makes the case that lattice-valued bitopology is categorically redundant. As a special application, traditional bitopology as represented by BiTop is (isomorphic in an extremely well-behaved way to) a strict subcategory of 4-Top, where 4 is the four element Boolean algebra; this makes the case that traditional bitopology is a special case of a much simpler fixed-basis topology.Support of Youngstown State University via a sabbatical for the 2005–2006 academic year is gratefully acknowledged.Rodabaugh, S. (2008). Functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics. Applied General Topology. 9(1):77-108. doi:10.4995/agt.2008.1871.SWORD771089

Politics, Principles and Pluralism: On why liberalism must be inconsistent if correct

In this dissertation, the author argues that constructivist foundations of political liberalism require a rarely recognized sort of pluralism—not only the familiar pluralism between ideas about how we ought to live that are the stock in trade of standard accounts of liberalism, but a pluralism about political foundations as well. The author argues that making sense of this requires revision to the way we sometimes understand key concepts (such as obligation), and develops an inconsistency-tolerant, pluralism friendly deontic logic for this purpose. A pluralist friendly obligation is argued to be one that represents moral and political principles in contrastive terms (analogous to contrastive explanation from Bas Van Fraassen), in virtue of the need to order acting upon prescriptions. The author develops a class of mathematical objects choices to model answers to why we should choose one policy over alternatives. Constructivist foundations also turn out to be prima facie pluralist foundations, in virtue of the nature of the norms guiding abstraction. This leads to a proof that, in a weakest base logic, legitimate moral or political codes in a pluralist context must reference each other. Upon explicating the distinction between perspectives that could consider unrealizable plans and perspectives that are themselves unrealizable, the author proves that in our world liberalism is itself an unrealizable plan. These results clearly illuminate what is at stake when justifying foundations for a liberal state

Fuzzy topological properties and hereditariness

Some known compactness notions and separation axioms already given in Chang-fuzzy topological spaces are extended to a more general context of categories where I -topological spaces on arbitrary I -sets are de3ned. The invariance under morphisms in these categories and the usual hereditary conditions of the considered topological properties in such a context are investigated