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The approximation of chainable continua by COTS
by
Richard G. Wilson
Universidad Autónoma Metropolitana, Unidad Iztapalapa, México D.F.
Coauthors: J. Kennedy (University of Delaware), R.D. Kopperman (City College, CUNY)
In [2], we have shown that every compact Hausdorff space X is the Hausdorff reflection of an inverse limit of finite T0-spaces. The finite spaces used for reconstructing X in this way are quotients of X and may indeed be rather complicated spaces. On the other hand, the connected ordered topological spaces (COTS) defined in [1] have a very simple structure and are used in digital topology as approximations to intervals of the real line.
It is well known that chainable continua can be characterized as being the inverse limits of sequences of unit intervals and since in their turn, these can be approximated (in the sense of [2]) by COTS, it is natural to ask whether chainable continua can also be approximated in this way. The aim of this talk is prove the following result:
Theorem. A topological space is a chainable continuum if and only if it is the Hausdorff reflection of an inverse limit of a sequence of finite COTS.
Bibliography
[1] E. Khalimsky, R. D. Kopperman and P. R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology and its Appl. 36 (1990), 1-17.
[2] Kopperman, R.D. and Wilson, R.G., Finite approximation of compact Hausdorff spaces, Topology Proceedings, Volume 22, 1999, 175-200.
Date received: May 27, 1999
Copyright © 1999 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 # cacl-26.