A new look at pointfree metrization theorems

summary:We present a unified treatment of pointfree metrization theorems based on an analysis of special properties of bases. It essentially covers all the facts concerning metrization from Engelking [1] which make pointfree sense. With one exception, where the generalization is shown to be false, all the theorems extend to the general pointfree context

σ\sigma-locales in Formal Topology

A $\sigma$-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. The aim of this paper is to show that $\sigma$-frames, actually $\sigma$-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology. Every $\sigma$-frame $L$ is the lattice of Lindel\"of elements (those for which each of their covers admits a countable subcover) of a formal topology of a specific kind which, in its turn, is a presentation of the free frame over $L$. We then give a constructive characterization of the smallest (strongly) dense $\sigma$-sublocale of a given $\sigma$-locale, thus providing a ``$\sigma$-version'' of a Boolean locale. Our development depends on the axiom of countable choice.Comment: Paper presented at the conference Continuity, Computability, Constructivity - From Logic to Algorithms (CCC 2017), Nancy, France, June 26-30 201

On Fixed-Point Free Selections and Multivalued Maps on R

[EN] We study fixed-point free multivalued selfmaps on R as well as continuous fixed-point free selection of multivalued selfmaps on R. We justify necessary conditions in some earlier results and motivate some new interesting questions.Burke, D.; Buzyakova, R. (2015). On Fixed-Point Free Selections and Multivalued Maps on R. Applied General Topology. 16(1):31-36. doi:http://dx.doi.org/10.4995/agt.2015.2291.SWORD3136161Aarts, J. M., Fokkink, R. J., & Vermeer, H. (1996). Variations on a theorem of lusternik and schnirelmann. Topology, 35(4), 1051-1056. doi:10.1016/0040-9383(95)00057-7Buzyakova, R. Z., & Chigogidze, A. (2011). Fixed-point free maps of Euclidean spaces. Fundamenta Mathematicae, 212(1), 1-16. doi:10.4064/fm212-1-1R. Engelking, General Topology, PWN, Warszawa, 1977. J. van Mill, The infinite-dimensional topology of function spaces, Elsevier, Amsterdam, 2001

Pointfree bispaces and pointfree bisubspaces

This thesis is concerned with the study of pointfree bispaces, and in particular with the pointfree notion of inclusion of bisubspaces. We mostly work in the context of d-frames. We study quotients of d-frames as pointfree analogues of the topological notion of bisubspace. We show that for every d-frame L there is a d-frame A(L) such that it plays the role of the assembly of a frame, in the sense that it has the analogue of the universal property of the assembly and that its spectrum is a bitopological version of the Skula space of the bispace dpt(L), the spectrum of L. Furthermore, we show that this bitopological version of the Skula space of dpt(L) is the coarsest topology in which the d-sober bisubspaces of dpt(L) are closed. We also show that there are two free constructions in the category of d-frames Act(L) and A_(L), such that they represent two variations of the bitopological version of the Skula topology. In particular, we show that in dpt(Act) the positive closed sets are exactly those d-sober subspaces of dpt(L) that are spectra of quotients coming from an increase in the con component, and that the negative closed ones are those that come from increases in the tot component. For dpt(A_(L)), we show that the positive closed sets are exactly those bisubspaces of dpt(L) that are spectra of quotients coming from a quotient of L+, and that the negative closed sets come in the same way from quotients of

Topological and metric emergence of continuous maps

We prove that the homeomorphisms of a compact manifold with dimension one have zero topological emergence, whereas in dimension greater than one the topological emergence of a C^0-generic conservative homeomorphism is maximal, equal to the dimension of the manifold. Moreover, we show that the metric emergence of continuous self-maps on compact metric spaces has the intermediate value property

Numerable open covers and representability of topological stacks

We prove that the class of numerable open covers of topological spaces is the smallest class that contains covers with pairwise disjoint elements and numerable covers with two elements, closed under composition and coarsening of covers. We apply this result to establish an analogue of the Brown--Gersten property for numerable open covers of topological spaces: a simplicial presheaf on the site of topological spaces satisfies the homotopy descent property for all numerable open covers if and only if it satisfies it for numerable covers with two elements and covers with pairwise disjoint elements. We also prove a strengthening of these results for manifolds, ensuring that covers with two elements can be taken to have a specific simple form. We apply these results to deduce a representability criterion for stacks on topological spaces similar to arXiv:1912.10544. We also use these results to establish new simple criteria for chain complexes of sheaves of abelian groups to satisfy the homotopy descent property.Comment: 25 pages. Comments and questions are welcome. Added strengthened results for the case of cartesian spaces (Theorems 1.3 and 1.7). v2: Identical to the journal version except for formatting and styl

Quantalic spectra of semirings

Spectrum constructions appear throughout mathematics as a way of constructing topological spaces from algebraic data. Given a localic semiring R (the pointfree analogue of a topological semiring), we define a spectrum of R which generalises the Stone spectrum of a distributive lattice, the Zariski spectrum of a commutative ring, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann–Lawson spectrum of a continuous frame. We then provide an explicit construction of this spectrum under conditions on R which are satisfied by our main examples. Our results are constructively valid and hence admit interpretation in any elementary topos with natural number object. For this reason the spectrum we construct should actually be a locale instead of a topological space. A simple modification to our construction gives rise to a quantic spectrum in the form of a commutative quantale. Such a quantale contains 'differential' information in addition to the purely topological information of the localic spectrum. In the case of a discrete ring, our construction produces the quantale of ideals. This prompts us to study the quantale of ideals in more detail. We discuss some results from abstract ideal theory in the setting of quantales and provide a tentative definition for what it might mean for a quantale to be nonsingular by analogy to commutative ring theory