Quotient topologies on topological semilattices

O.Hryniv

It is proven that for a closed $\sigma$-compact ideal $I$ in a locally compact (Lawson) semilattice $S$ the quotient $S/I$ is a topological (Lawson) semilattice. Also we construct several counterexamples showing that the above result cannot be improved. One is an example of a countable subsemilattice $S\subset \mathbb R^4$ containing a closed ideal $I\subset S$ such that $S/I$ fails to be a topological semilattice. The other is an example of a metrizable locally compact locally countable Lawson semilattice $S$ of size $|S|=\aleph_1$ containing a closed discrete ideal $I$ such that the quotient $S/I$ fails to be a topological semilattice. Moreover, the quotient topology on $S/I$ in the category of topological semilattices differs from the quotient topology in the category of Lawson semilattices. This answers in negative a question of J.Lawson and B.Madison.

1. Blass A. Combinatorial cardinal characteristics of the continuum. in: M. Foreman, A.~Kanamori, M.~Magidor (eds.). Handbook of set theory, Kluwer (to appear).

2. Carruth J.H., Hildebrant J.A., Koch R.J. The theory of topological semigroups, Marsel Dekker, 1983.

3. van Douwen E. The integers and topology, in: K.Kunen, J. Vaughan (eds.) Handbook of set-theoretic topology, North-Holland, Amsterdam-New York, 1984, 111--167.

4. Энгелькинг Р. Общая топология. -- М.: Мир, 1986.

5. González G., Closed congruences on semigroups, Divulgaciones Matem\'aticas 9:1 (2001), 103--107.

6. Lawson J., Topological semilattices with small subsemilattices, J. London Math. Soc. ( 2):1 (1969), 719--724.

7. Lawson J., Madison B. On Congruences and Cones, Math. Zeitschrift 120 (1971), 18--24.

8. Vaughan J., Small cardinals and topology, in: J.van Mill, G.M.Reed (eds.) Open problems in topology. (Elsevier Sci. Publ. 1990), 197--216.

	Quotient topologies on topological semilattices
Author	O.Hryniv hryniv@ukr.net Ivan Franko Lviv National University, Faculty of Mechanics and Mathematics
Abstract	It is proven that for a closed $\sigma$-compact ideal $I$ in a locally compact (Lawson) semilattice $S$ the quotient $S/I$ is a topological (Lawson) semilattice. Also we construct several counterexamples showing that the above result cannot be improved. One is an example of a countable subsemilattice $S\subset \mathbb R^4$ containing a closed ideal $I\subset S$ such that $S/I$ fails to be a topological semilattice. The other is an example of a metrizable locally compact locally countable Lawson semilattice $S$ of size $\|S\|=\aleph_1$ containing a closed discrete ideal $I$ such that the quotient $S/I$ fails to be a topological semilattice. Moreover, the quotient topology on $S/I$ in the category of topological semilattices differs from the quotient topology in the category of Lawson semilattices. This answers in negative a question of J.Lawson and B.Madison.
Keywords	compact ideal, Lawson semilatttice, quotient
DOI	doi:10.30970/ms.23.2.136-142
Reference	1. Blass A. Combinatorial cardinal characteristics of the continuum. in: M. Foreman, A.~Kanamori, M.~Magidor (eds.). Handbook of set theory, Kluwer (to appear). 2. Carruth J.H., Hildebrant J.A., Koch R.J. The theory of topological semigroups, Marsel Dekker, 1983. 3. van Douwen E. The integers and topology, in: K.Kunen, J. Vaughan (eds.) Handbook of set-theoretic topology, North-Holland, Amsterdam-New York, 1984, 111--167. 4. Энгелькинг Р. Общая топология. -- М.: Мир, 1986. 5. González G., Closed congruences on semigroups, Divulgaciones Matem\'aticas 9:1 (2001), 103--107. 6. Lawson J., Topological semilattices with small subsemilattices, J. London Math. Soc. ( 2):1 (1969), 719--724. 7. Lawson J., Madison B. On Congruences and Cones, Math. Zeitschrift 120 (1971), 18--24. 8. Vaughan J., Small cardinals and topology, in: J.van Mill, G.M.Reed (eds.) Open problems in topology. (Elsevier Sci. Publ. 1990), 197--216.
Pages	136-142
Volume	23
Issue	2
Year	2005
Journal	Matematychni Studii
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