On Dimension of Bitopological Spaces
by
M. Jelic

Definition: Let (X, t₁, t₂) be a bitopological space (briefly bispace X).

t₁t₂dmX = -1 if X = \emptyset.

t₁t₂dmX = 0 if every finite t₁-open cover of X has a finite t₂ -open refinement whose nerve is totaly unordered.

t₁t₂dmX <= n, n = 1, 2, ... if every finite t₁-open cover U of X has a finite t₂-open V refinement such that dsN(V) <= n+1.

t₁t₂dmX = n if t₁t₂dmX <= n and t₁t₂dmX \not <= n-1.

t₁t₂dmX = +\infty if t₁t₂dmX \not <= n for n=-1, 0, 1, ... .

The next propositions contain useful characterizations of this dimension.

Proposition 1 Let X = A \cup B be a bispace with t₂-closed set A in X. Let (X, t₂) be a normal space and t₁t₂dmA <= m, t₁t₂dmB <= n, m <= n, m, n = 1, 2, ... and let each t₂-open cover of X have a t₁-open refinement. Then t₁t₂dmX <= m+n+1.

Proposition 2 Let t₁t₂dmG <= n for every t₁-open set G of a bispace X. Then t₁t₂dmA <= n for each A subset or equal X.

Proposition 3 Let X = A \cup B, where (X, t₁) and (X, t₂) are normal spaces. Let A be t₂-closed in X and t₂t₁dmA <= n (n >= 0) and t₁dmB = 0. Then t₁t₂dmX <= n+2.

Date received: June 24, 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 # caah-46.