Related papers: Sober topological spaces valued in a quantale
We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of…
Completeness for a (topological) space is often based on the existence of special structures (such as metrics, uniformities, proximities, convergences, etc) that explicitly induce the topology, making the completeness induction-dependent.…
With this paper we hope to contribute to the theory of quantales and quantale-like structures. It considers the notion of $Q$-sup-algebra and shows a representation theorem for such structures generalizing the well-known representation…
The aim of this paper is to study the topological properties of some classes of subsemimodules endowed with a subbasis closed-set topology. We show that such spaces are $T_0$. When the semimodule is finitely generated, those spaces are…
Notions of topological free entropy and of free capacity are introduced in the $C^*$-algebra context. Basic properties, basic problems and connections to potential theory and random matrix theory are discussed.
We investigate the question whether a given homogeneous ideal is a limit of saturated ones. We provide cohomological necessary criteria for this to hold and apply them to a range of examples. Our motivation comes from the theory of border…
Spaces of quasi-invariant measures supplied with different topologies are studied. Their embeddings, projective decompositions, conditions for their metrizability are investigated. Theorems about convergence of nets of quasi-invariant…
The interplay between the two fundamental concepts of topological order and reflection positivity allows one to characterize the ground states of certain many-body Hamiltonians. We define topological order in an appropriate fashion and show…
A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections.…
It is shown that the duals of several categories of topological flavour, like the categories of ordered sets, generalised metric spaces, probabilistic metric spaces, topological spaces, approach spaces, are quasivarieties, presenting a…
We introduce the class of sober rings and investigate it through several key results, highlighting connections to some other known classes of rings. We analyze sufficient conditions for a ring to be sober, as well as necessary conditions.…
Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes…
The notion of computability is stable (i.e. independent of the choice of an indexing) over infinite-dimensional vector spaces provided they have a finite "tensorial dimension". Such vector spaces with a finite tensorial dimension permit to…
We study semi--algebraic domains associated with symplectic tori and conjecturally identified with spaces of stability conditions on the Fukaya categories of these tori. Our motivation is to test which results from the theory of flat…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
The role of topology in elementary quantum physics is discussed in detail. It is argued that attributes of classical spatial topology emerge from properties of state vectors with suitably smooth time evolution. Equivalently, they emerge…
In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces $X$ and $Y$, it is proved that $Y$ is H-sober iff the function space $\mathbb{C}(X, Y)$ of all continuous…
At present, our notion of space is a classical concept. Taking the point of view that quantum theory is more fundamental than classical physics, and that space should be given a purely quantum definition, we revisit the notion of Euclidean…
For evolution of flat universe, we classify late time and future attractors with scaling behavior of scalar field quintessence in the case of potential, which, at definite values of its parameters and initial data, corresponds to exact…
This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The…