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Notes on Locally Compact Connected Topological Lattices

Published online by Cambridge University Press:  20 November 2018

Tae Ho Choe
Tae Ho Choe*
Affiliation:
McMaster University, Hamilton, Ontario
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I t was shown in (2) that if

(1) L is a locally compact connected topological lattice and if

(2) L is topologically contained in R2, the Euclidean plane, then each compact subset of L has an upper bound and a lower bound in L. I t was also asked whether this result could be proved without assuming condition (2). In this note, we show that this result continues to hold if condition (2) is weakened to: L is finite-dimensional.

In (11), it was shown that the centre of a compact topological lattice is totally disconnected. We shall prove t h a t this result is also true even in a locally compact, locally convex topological lattice with 0 and 1. This yields that any locally compact topological Boolean algebra is totally disconnected.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1533 - 1536
Copyright
Copyright © Canadian Mathematical Society 1969

References

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