We present some results about the class of Alexandroff topologies (i.e. topologies where the intersection of arbitrary many open sets is open) from the perspective obtained when they are viewed as closed subsets of the Cantor cuber 2 (the power set of X with the product topology).

In this paper we consider linear dynamic equations on Cantor ternary set. We derive an explicit formula for the solution of the associated initial value problem. In addition, we consider a special case in which the solution in question is given by a rather simple formula.

In this work, we develop shape expansions of minimal matchbox manifolds without holonomy, in terms of branched manifolds formed from their leaves. Our approach is based on the method of coding the holonomy groups for the foliated spaces, to define leafwise regions which are transversely stable and are adapted to the foliation dynamics. Approximations are obtained by collapsing appropriately cho...

The space of one-sided infinite words plays a crucial rôle in several parts of Theoretical Computer Science. Usually, it is convenient to regard this space as a metric space, the Cantor-space. It turned out that for several purposes topologies other than the one of the Cantor-space are useful, e.g. for studying fragments of first-order logic over infinite words or for a topological characterisa...

The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholt...

We consider the Complex Stone-Weierstrass Property (CSWP), which is the complex version of the Stone-Weierstrass Theorem. If X is a compact subspace of a product of three linearly ordered spaces, then X has the CSWP if and only if X has no subspace homeomorphic to the Cantor set. In addition, every finite power of the double arrow space has the CSWP. These results are proved using some results ...

For 20th century mathematicians, the role of Cantor's sets has been that of the ideally featureless canvases on which all needed algebraic and geometrical structures can be painted. (Certain passages in Cantor's writings refer to this role.) Clearly, the resulting contradiction, 'the points of such sets are distinct yet indistinguishable', should not lead to inconsistency. Indeed, the productiv...

The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper.

It has been recently proved that analytic capacity, γ, is semiadditive. This result is a consequence of the comparability between γ and γ+, a version of γ originated by bounded Cauchy potentials of positive measures. In this paper we describe the main ideas involved in the proof of this result and we give a complete proof of it in the particular case of the N -th approximation of the corner qua...