Reminder: Today's Seminar by Andrej Bauer at 3:30

[Margery Ishmael]

LOGIC SEMINAR AND COMPUTER SCIENCE SEMINAR

DATE: Thursday, Nov. 16, 2000
TIME:   3:30-5:00
PLACE: Ryerson (annex) 277 - followed by refreshments in Ryerson 255
TITLE: Computability for Topological Spaces
SPEAKER: Andrej Bauer
School of Computer Science
Carnegie Mellon University
Pittsburgh

ABSTRACT:
We will present a framework for computable topology based on recent
developments that began in 1996 with Dana Scott's equilogical spaces.
The starting points are two observations about the powerset of the
natural numbers P(N), equipped with the Scott topology: first, the
subspaces of P(N) are precisely all the countably based T_0-spaces;
second, P(N) is equipped with a natural notion of computability,
namely recursive enumerability. This leads to a formulation of
computable topology on all countably based T_0-spaces, with computable
maps between them that are defined on the non-computable as well as
the computable points. This setup can then be extended to equilogical
spaces by equipping the topological spaces with equivalence
relations. The benefit of adding the equivalence relations is that
they let us distinguish between the points of a space and the names of
points, similar to the distinction between r.e. sets and Goedel codes
of r.e. sets.  Equilogical spaces form a category that has very good closure
properties. For example, it is cartesian closed, which means that we
can form all function spaces, function spaces of function spaces, etc.
This cannot be done if we restrict attention only to topological
spaces. In the last part of the talk we will show how computability on
real numbers and real functions is treated in equilogical spaces.

Visitor's Host: Prof. Robert I. Soare




====================================================
Margery Ishmael
Department of Computer Science
1100 E. 58th St.
Chicago, IL 60637

tel: 773 834-8977  fax: 773 702-8487

marge at cs.uchicago.edu 
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