Related papers: A bump in the road in elementary topology
We consider the problem of constructing a weakly-continuous mapping extending continuous mapping defined on a dense set of a topological space to the entire space. Theorem on necessary and sufficient conditions for the existence of such an…
We construct a category that classifies compact Hausdorff spaces by their shape and finite topological spaces by their weak homotopy type.
We characterize geodesic paths in the $n$-dimensional unit sphere under sup norm. A geodesic path between two points is a shortest curve joining the two points.
Questions such as whether we live in a spatially finite universe, and what its shape and size may be, are among the fundamental open problems that high precision modern cosmology needs to resolve. These questions go beyond the scope of…
Coincidences of maps between smooth manifolds are studied via a geometric approach which involves (nonstabilized) normal bordism theory and pathspaces.
A function on a topological space is called unimodal if all of its super-level sets are contractible. A minimal unimodal decomposition of a function $f$ is the smallest number of unimodal functions that sum up to $f$. The problem of…
We want to investigate 'spaces' where paths have a 'weight', or 'cost', expressing length, duration, price, energy, etc. The weight function is not assumed to be invariant up to path-reversion. Thus, 'weighted algebraic topology' can be…
We introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site $({\mathcal{C}}, J)$ and that of…
Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is…
The concept of relative sectional category expands upon classical sectional category theory by incorporating the pullback of a fibration along a map. Our paper aims not only to explore this extension but also to thoroughly investigate its…
A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and…
Topological entropy has been one of the most difficult to implement of all the entropy-theoretic notions. This is primarily due to finite sample effects and high-dimensionality problems. In particular, topological entropy has been…
This brief note gives a survey on results relating to existence of closed points on schemes, including an elementary topological characterization of the schemes with (at least one) closed point.
We introduce a notion of weak convergence in arbitrary metric spaces. Metric functionals are key in our analysis: weak convergence of sequences in a given metric space is tested against all the metric functionals defined on said space. When…
In this paper, we introduce a thinness in sense to a type of relative capacity for weighted variable exponent Sobolev space. Moreover, we reveal some properties of this thinness and consider the relationship with finely open and finely…
Topological deep learning is a formalism that is aimed at introducing topological language to deep learning for the purpose of utilizing the minimal mathematical structures to formalize problems that arise in a generic deep learning…
Digital topology has its own working conditions and sometimes differs from the normal topology. In the area of topological robotics, we have important counterexamples in this study to emphasize this red line between a digital image and a…
Regular polygons are characterized as area-constrained critical points of the perimeter functional with respect to particular families of perturbations in the class of polygons with a fixed number of sides. We also review recent results in…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
Realistic metric spaces (such as road/transportation networks) tend to be much more algorithmically tractable than general metrics. In an attempt to formalize this intuition, Abraham et~al.\ (SODA 2010, JACM 2016) introduced the notion of…