Atlas: Topology-Generated Adjacency Relations on $Z^n$, and a Characterization of Finite Products of Khalimsky Lines by T. Y. Kong

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1999 Summer Conference on Topology and its Applications
August 4-7, 1999
C.W. Post Campus of Long Island University
Brookville, NY, USA

Organizers
Sheldon Rothman, Ralph Kopperman

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Topology-Generated Adjacency Relations on Zⁿ, and a Characterization of Finite Products of Khalimsky Lines
by
T. Y. Kong
Queens College, CUNY

It is well known that while there is a topology on Z² for which the connected sets are exactly the 4-connected subsets of Z², there is no topology for which the connected sets are exactly the 8-connected subsets of Z². We address the following more general question: For which symmetric binary relations \alpha on Zⁿ does there exist a topology \tau_\alpha on Zⁿ such that the \tau_\alpha-connected sets are exactly the \alpha-connected subsets of Zⁿ? (A set S is said to be \alpha-connected iff x \alpha_S^* y for all x, y in S, where \alpha_S^* is the reflexive transitive closure of the restriction of \alpha to S.)

If such a topology \tau_\alpha exists then we say that the relation \alpha is topology-generated, and say that \tau_\alpha generates \alpha. We present results that can be used to find, for any positive integer n, all topology-generated symmetric binary relations \alpha on Zⁿ that satisfy the following three conditions:

For all x, y in Zⁿ, ||x - y||₁ = 1 ===> x \alpha y

For all x, y in Zⁿ, x \alpha y ===> ||x - y||_\infty = 1

For all x in Zⁿ, the set {x} \cup {y | x \alpha y} is l₁-connected, where l₁ is the binary relation on Zⁿ defined by x l₁ y <===> ||x - y||₁ = 1.

The problem of finding all topology-generated relations that satisfy these conditions was essentially solved for n <= 3 by Eckhardt and Latecki. We use a different approach to attack the problem for all positive integers n.

For i = 0, 1 we call the topology on Z that is generated by {{j-1, j, j+1} | j \equiv i mod 2} a Khalimsky line topology. For all positive integers n, the only simply connected topologies on Zⁿ which generate a relation that satisfies conditions 1 and 2 above are the products of n Khalimsky line topologies.

Date received: June 10, 1999