Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory

In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters

On the injectivity of Boolean algebras

summary:The functor taking global elements of Boolean algebras in the topos \text{\bold{Sh}\frak B} of sheaves on a complete Boolean algebra $\frak B$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $\frak B$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts

Epimorphisms and maximal covers in categories of compact spaces

[EN] The category C is "projective complete"if each object has a projective cover (which is then a maximal cover). This property inherits from C to an epireflective full subcategory R provided the epimorphisms in R are also epi in C. When this condition fails, there still may be some maximal covers in R. The main point of this paper is illustration of this in compact Hausdorff spaces with a class of examples, each providing quite strange epimorphisms and maximal covers. These examples are then dualized to a category of algebras providing likewise strange monics and maximal essential extensions.Banaschewski, B.; Hager, A. (2013). Epimorphisms and maximal covers in categories of compact spaces. Applied General Topology. 14(1):41-52. doi:10.4995/agt.2013.1616.SWORD4152141H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1966) 214–249.R. Engelking, General Topology, Heldermann 1989.A. Gleason, Projective topological spaces, Ill. J. Math. 2 (1958), 482–489.L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag 1976.HAGER, A. W. (1989). Minimal Covers of Topological Spaces. Annals of the New York Academy of Sciences, 552(1 Papers on Gen), 44-59. doi:10.1111/j.1749-6632.1989.tb22385.xHager, A. W., & Martinez, J. (1998). Singular Archimedean lattice-ordered groups. Algebra Universalis, 40(2), 119-147. doi:10.1007/s000120050086A. Hager and L. Robertson, Representing and ringifying a Riesz space, Symp. Math. XXI (1977), 411–431.H. Herrlich and G. Strecker, Category Theory, Allyn and Bacon 1973.Kennison, J. F. (1965). Reflective functors in general topology and elsewhere. Transactions of the American Mathematical Society, 118, 303-303. doi:10.1090/s0002-9947-1965-0174611-9Porter, J. R., & Woods, R. G. (1988). Extensions and Absolutes of Hausdorff Spaces. doi:10.1007/978-1-4612-3712-9V. Trnková, Non-constant continuous mappings of metric or compact Hausdorff spaces, Comm. Math. Univ. Carol. 13 (1972), 283–295

The Potential Role of Cathelicidin Peptides for the Treatment of Lower Respiratory Tract Infections

Bacterial pneumonia, one of the leading causes of hospitalization and mortality worldwide, is caused by the colonization of invasive bacteria in the airways, leading to pulmonary inflammation and lung dysfunction. The development of antibiotic resistant bacterial infections has limited the effectiveness of current therapeutics and is particularly concerning in the setting of chronic bacterial infections, such as observed in cystic fibrosis and ventilator-associated pneumonia. The development of novel therapeutics for the treatment of multi-drug resistant bacterial pneumonia is urgently needed. The overall objective of this body of work was the development of a new therapeutic compound to treat multi-drug resistant bacterial infections in the lung. It was hypothesized that a cathelicidin/exogenous surfactant compound could be developed as a novel therapeutic agent for the treatment of bacterial pneumonia. To test this hypothesis, we first screened several cathelicidin peptides combined with a commercial exogenous surfactant, bovine lipid extract surfactant (BLES), and identified a lead compound (CATH-2) for further testing. Following this, we investigated three main outcomes: 1) the antimicrobial activity of CATH-2 and BLES+CATH-2 against multi-drug resistant, clinically isolated bacteria; 2) the immunomodulatory potential of both CATH-2 and BLES+CATH-2 in vivo; and 3) the bactericidal activity of BLES+CATH-2 treatment in vivo, using two models of bacterial pneumonia. It was discovered that CATH-2 was able to kill bacteria in vitro. In addition, CATH-2 killed bacteria administered into the lungs of mice did not induce an inflammatory response in vivo, and that the ability to prevent this inflammation was maintained by BLES+CATH-2. Finally, while BLES+CATH-2 is able to kill multi-drug resistant, clinically derived bacteria in vitro, there is little bactericidal activity of BLES+CATH-2 in in vivo models of bacterial pneumonia. Overall, we identified the therapeutic potential of BLES+CATH-2 for use as a therapeutic treatment for bacterial pneumonia. Despite promising in vitro activity, we were unable to show bactericidal activity for BLES+CATH-2 in vivo. Future directions will require the optimization of the surfactant-cathelicidin compound in order to develop a viable therapeutic for clinical practice

Ring ideals and the Stone-Čech compactification in pointfree topology

AbstractThis paper shows that the compact completely regular coreflection in the category of frames is given by the frame of Jacobson radical ideals of the ring RL of real-valued continuous functions on L, as an alternative to its familiar representations in terms of (i) the l-ideals of RL as lattice-ordered ring or (ii) the ideals of the bounded part of RL which are closed in the usual uniform topology. Further, in analogy with this, the compact zero-dimensional coreflection will also be described in terms of ring ideals, this time of the ring ZL of integer-valued continuous functions on L

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

A globalisation of the Gelfand duality theorem

AbstractIn this paper we bring together results from a series of previous papers to prove the constructive version of the Gelfand duality theorem in any Grothendieck topos E, obtaining a dual equivalence between the category of commutative C∗-algebras and the category of compact, completely regular locales in the topos E