Hernandez-Hernandez, Fernando

General Topology. Topology is a kind of qualitative geometry in which one is interested not in quantitative notions such as the traditional geometry was at its beginnings but in those properties that a "space" posses and that remain after a continuous deformation of it. Some people has associated the starting point of topology back to the time of Euler and the Königsberg bridge problem. However, it seems to me that defining what means that a given point is in the closure of a given set is the beggining of the true topology going away from geometry. A. V. Arkhange'skii once said that topology is "the science of infinite closeness without distance". I believe that this really describes the essence of topology. Topology is the study of sets on which one has a notion of "closeness" —enough to decide which functions defined on it are continuous. Topology is used in nearly all areas of mathematics in one form or another; so there are many braches of topology. Set-theoretic topology is the branch which studies the most abstract forms of spaces and it is characterized by using wealth of tools from set theory. This is the kind of topology I am interested in.

Set Theory. It was initiated by Georg Cantor. The problems about cardinals were the main topic in a first stage of set theory. In that initial period, every "conceivable" set was thought to exists, every collection for which it was possible to say in some way what its elements were was considered a set. It soon turned out that this viewpoint is untenable. The development of axiomatic set theory was then a need. It came the second stage of set theory in which the search for the better axiom system for set theory was the main topic. Enriched and fortified by axioms, results and techniques axiomatic set theory was launched on its independent course by Gödel in the 1930's. In the early 1960's set theory was transformed due largely to the creation of forcing by Cohen. According to D. S. Scott "Set theory could never be the same after Cohen, and there is simply no comparison whatsoever in the sophistication of our knowledge about models of set theory today as contrasted to the pre-Cohen era". It is not at all surprising that this development of set theory has a deep impact in topology. Exploring, learning, and taking advantage of it is what moves my interest in set theory.

Fernando Hernández-Hernández

Set Theory The Jean A. Larson's list of homepages of set theorists.

Shelah's Archive

The Hompege of the Axiom of Choice

arXiv.org e-Print archive

Mathematicians who were born or died on this day

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