ILCLI Short Course

Unit of Formal Methods for Language, Cognition, and Action

  • July 6 &7, 11 a.m.-2:00 p.m.
  • July 13 &14, 11 a.m.-2:00 p.m
  • E. Ezenarro (ILCLI), J. M. Larrazabal (ILCLI) and J. Martínez (Univ. Barcelona & Ikerbasque-ILCLI)

A Short Introduction to Space and Measure

Venue: ILCLI Seminar Room

This is a self-contained short course in general topology, with the purpose of presenting the basic notions, properties and methods in the study of topological spaces and metric spaces, stressing on easy examples and applications in mathematics and logic. For this survey the fundamentals of naïve set theory are required, and also a basic knowledge of real analysis, as well as knowledge of the semantics of first-order logic and propositional modal logic.

PEOPLE interested in taking part send a message by July 5, 2011, to: jesusmaria.larrazabal@ehu.es

Topics

  1. Topological Spaces. Basic definitions. Hausdorff spaces. Continuous functions. Homeomorphisms.
  2. Products of topological spaces. Connected Spaces.
  3. The notion of 'measure'. Scales measurement. Metric spaces.
  4. Normal spaces. Normed spaces. Banach spaces. Hilbert spaces.
  5. Compact spaces. Compact Hausdorff spaces. Compact metric spaces.
  6. Compacity in logic. Topology for modal logic.

Main References

  • N. Bourbaki, General Topology. Part 1. Reading: Addison-Wesley, 1966.
  • J. Dixmier, General Topology. New York: Springer, 1984.
  • P. Halmos, Naïve Set Theory. Princeton: Van Nostrand, 1960.
  • F. Hausdorff, Set Theory. New York: Chelsea, 1957.
  • I. M. James (ed.), History of Topology. Amsterdam: North-Holland/Elsevier, 1999.
  • J. Kelley, General Topology. Princeton: Van Nostrand, 1955.
  • K. Kuratowski, Introduction to Set Theory and Topology. Oxford: Pergamon Press, 1961.
  • B. Mendelson, Introduction to Topology. New York: Dover, 1962.
  • H. Poincaré, Analysis Situs (1895), in: Oeuvres, Vol. VI. Paris: Gauthier-Villars, 11 volumes, 1916-1956.
  • W. Sierpinski, General Topology. Toronto: Toronto University Press, 1934.
  • W. A. Sutherland, Introduction to Metric and Topological Spaces. Oxford: Oxford University Press, 1975.
  • S. Willard, General Topology. Reading: Addison-Wesley, 1970.