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We show some basic results on the characterization of quasi-Polish spaces in terms of spaces of ideals, with an emphasis on the connections with computable topology.
In this paper, the author introduce and study the notion of pre-{\gamma}-I-open sets in ideal topological space.
The lattice of ideals of the torsion ideal of the endomorphism ring of an abelian p-group is classified by a system of cardinal invariants.
The interrelations between various classes of convergence spaces defined by countability conditions are studied. Remarkably, they all find characterizations in the usual space of ultrafilters in terms of classical topological properties.…
Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the…
Crystallographic tilings of the Euclidean space $\mathbb{E}^n$ are defined as simple tilings whose group of isometric automorphisms is crystallographic. To classify crystallographic tilings by their automorphism groups it is necessary to…
We define analytic indices which involve the eta form and the analytic torsion form. We show that these indices are independent of the geometric choices made in their definitions, and hence are topological in nature.
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack…
We say that an ideal I is homogeneous, if its restriction to any I-positive subset is isomorphic to I. The paper investigates basic properties of this notion -- we give examples of homogeneous ideals and present some applications to…
We introduce and prove the consistency of a new set theoretic axiom we call the \emph{Invariant Ideal Axiom}. The axiom enables us to provide (consistently) a full topological classification of countable sequential groups, as well as fully…
We consider \Gamma-equivariant principal G-bundles over proper \Gamma-CW-complexes with prescribed family of local representations. We construct and analyze their classifying spaces for locally compact, second countable topological groups…
On objects of a triangulated category with a stability condition, we construct a topology.
For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the…
We introduce and study the proper topological complexity of a given configuration space, a version of the classical invariant for which we require that the algorithm controlling the motion is able to avoid any possible choice of ``unsafe''…
We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra $\mathcal{NT}(X)$ of a product system $(A, X)$ over $\mathbb{N}^n$. We obtain a clear description of $X$-invariant ideals in $A$, that is, restrictions of…
We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard generic monomial ideals to find invariants for rational maps of spheres and…
In this paper, we discuss the classification problem for linear time-invariant multivariable systems without control. It turns out that the observability and stability are invariant for topological equivalent systems. Abstract results…
This overview article makes the case for how topological concepts can enrich research in machine learning. Using the Euler Characteristic Transform (ECT), a geometrical-topological invariant, as a running example, I present different use…
In this paper we have discussed the ideals in Abel Grassmann's groupoids and construct their topologies.