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SYSTEMS OF ISLANDS WITH CONTINUOUS HEIGHT FUNCTIONS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 94 / Issue 3 / June 2013
- Published online by Cambridge University Press:
- 01 May 2013, pp. 385-396
- Print publication:
- June 2013
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- Article
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We investigate island systems with continuous height functions and strongly laminar systems which are laminar systems containing sets with disjoint boundaries. In the discrete case, we show that for a maximal rectangular system of islands
$ \mathcal{H} $ on an 
$m$ by 
$n$ rectangular grid we have 
$\lceil \min (m, n)/ 4\rceil \leq \vert \mathcal{H} \vert \leq \lceil m/ 2\rceil \lceil n/ 2\rceil $. In the continuous case we show that under some conditions maximal strongly laminar systems 
$ \mathcal{H} $ have cardinality 
${\aleph }_{0} $ or 
${2}^{{\aleph }_{0} } $ and present examples with 
$\vert \mathcal{H} \vert = {\aleph }_{0} $.
