In this paper, our purpose is twofold. Firstly, the tensor andresiduum operations on $L-$nested systems are introduced under thecondition of complete residuated lattice. Then we show that$L-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. Thus the new representation theorem of$L-$subsets on complete re...

We show that a large class of Cantor-like sets of Rd, d ≥ 1, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when d ≥ 2. An analogous result is also proved for subsets of Rd arising in the study of geodesic flows corresponding to (d+1)-dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable...

We present a formalization in ACL2(r) of three proofs originally done by Cantor. The first two are different proofs of the nondenumerability of the reals. The first, which was described by Cantor in 1874, relies on the completeness of the real numbers, in the form that any infinite chain of closed, bounded intervals has a non-empty intersection. The second proof uses Cantor’s celebrated diagona...

‎In this article we introduce $mu$-filtered fuzzy module with a family of fuzzy submodules.  It shows the relation between $mu$-filtered fuzzy modules and crisp filtered modules by level sets. We investigate fuzzy topology on the $mu$-filtered fuzzy module and apply that to introduce fuzzy completion. Finally we extend Krull's intersection theorem of fuzzy ideals by using concept $mu$-adic comp...

When A and B are countable dense subsets of R, it is a well-known result of Cantor that A and B are order-isomorphic. A theorem of K.F. Barth and W.J. Schneider states that the order-isomorphism can be taken to be very smooth, in fact the restriction to R of an entire function. J.E. Baumgartner showed that consistently 2א0 > א1 and any two subsets of R having א1 points in every interval are ord...

For a very simple family of self-similar sets with two pieces, we prove, using a technique of Solomyak, that the intersection of the pieces can be a Cantor set with any dimension in [0, 0.2] as well as a finite set of any cardinality 2m. The main point is that the open set condition is fulfilled for all these examples.

Eighty years ago, Felix Hausdorff and Paul Alexandroff published independently a theorem asserting that every compact metric space is a continuous image of the Cantor set. This theorem found its application in various branches of mathematics and played also an important role in the theory of curves. The complete characterization of continuous interval images (i.e. Jordan curves) given by the Ha...