Set theory and topology. An introduction to the foundations of analysis. Part II: Topology - Fundamental notions

Set theory and topology. An introduction to the foundations of analysis. Part I: Sets, relations, numbers

We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of number systems.

Foundations of Mathematics

This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Godel's proof, the first modern incompleteness theorem, Turing's halting problem, and a piece of postmodern metamathematics, the halting probability Omega. The second part looks forward to the new century and discusses the convergence of theoretical physics and theoretical computer science and hopes for a theoretical biology, in which the notions of algorithm and information are again crucial.

Topology and biology: From Aristotle to Thom

Ren{é} Thom discovered several refined topological notions in the writings of Aristotle, especially the biological. More generally, he considered that some of the assertions of the Greek philosophers have a definite topological content, even if they were written 2400 years before the field of topology was born. In this article, we expand on these ideas of Thom. At the same time, we highlight the importance of biology in the works of Aristotle and Thom and we report on their conception of mathematics and more generally on science. The final version of this article will appear in the book \emph{…

The notion of space in Grothendieck: from schemes to a geometry of forms

In this essay we give a general picture about the evolution of Grohendieck's ideas regarding the notion of space. Starting with his fundamental work in algebraic geometry, where he introduces schemes and toposes as generalizations of classical notions of spaces, passing through tame topology and ending with the formulation of a geometry of forms, we show how the ideas of Grothendieck evolved from pure mathematical considerations to physical and philosophical questions about the nature and structure of space and its mathematical models.

马尔可夫可达拓扑及其概率结构

The Markov Accessibility Topology and its Probabilistic Structure

本文将典范拓扑与离散时间马尔可夫链的转移算子关联，定义了马尔可夫可达拓扑，建立了概率可达性与序理论拓扑的联系，还刻画了吸收子集等概念并研究了通信商与可 lump 性的关系。