Basic Properties of Metrizable Topological Spaces

We continue Mizar formalization of general topology according to the book [11] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that metrizable topological space. We also define Lindel¨of spaces and state the above theorem in this special case. We also introduce the concept of separation among two subsets (see [12]).Institute of Computer Science, University of Białystok, PolandGrzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Józef Białas and Yatsuka Nakamura. The theorem of Weierstrass. Formalized Mathematics, 5(3):353-359, 1996.Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481-485, 1991.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Ryszard Engelking. General Topology, volume 60 of Monografie Matematyczne. PWN-Polish Scientific Publishers, Warsaw, 1977.Ryszard Engelking. Teoria wymiaru. PWN, 1981.Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55-59, 1999.Adam Grabowski. On the Borel families of subsets of topological spaces. Formalized Mathematics, 13(4):453-461, 2005.Adam Grabowski. On the boundary and derivative of a set. Formalized Mathematics, 13(1):139-146, 2005.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Zbigniew Karno. Maximal discrete subspaces of almost discrete topological spaces. Formalized Mathematics, 4(1):125-135, 1993.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233-236, 1996.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Can a computer be "pushed" to perform faster-than-light?

We propose to "boost" the speed of communication and computation by immersing the computing environment into a medium whose index of refraction is smaller than one, thereby trespassing the speed-of-light barrier.Comment: 7 pages, 1 figure, presented at the UC10 Hypercomputation Workshop "HyperNet 10" at The University of Tokyo on June 22, 201

The discrete fragmentation equations : semigroups, compactness and asynchronous exponential growth

In this paper we present a class of fragmentation semigroups which are compact in a scale of spaces defined in terms of finite higher moments. We use this compactness result to analyse the long time behaviour of such semigroups and, in particular, to prove that they have the asynchronous growth property. We note that, despite compactness, this growth property is not automatic as the fragmentation semigroups are not irreducible