Many-Valued Functions and Fixed Points in Digital Topology
by
M.B. Smyth
Imperial College

Under the heading "digital topology" we include the (standard) graph-theoretic material of Rosenfeld and others, the approach via Khalimsky spaces, and - more unusually perhaps - the "tolerance geometry" of Zeeman. We argued in [1] that a framework, within which these approaches can be studied in a unified manner along with ördinary" topology, is available in the form of a suitable category of generalized topological structures, akin to Cech closure spaces. In that context, for example, ordinary spaces arise as inverse limits of digital spaces. We turn now to digital functions. To have a sufficiently flexible apparatus (in particular, for the approximation of ördinary" functions), these have to be taken as many-valued. We investigate fixed points. As it happens, "digital" analogues of Brouwer's Theorem have been rediscovered a number of times. Since we work with many-valued functions, however, we have to develop analogues of Kakutani's Theorem or related results. In summary, we address the following topics, of which (2) and (3) involve new material: (1) Ordinary spaces as inverse limits of digital spaces (2) Approximation of ordinary functions by many-valued digital functions (3) Fixed points for many-valued digital functions.

[1]

M.Smyth, Semi-metrics, closure spaces and digital topology. TCS (in press)

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-78.