Families of mutually complementary topologies

Anderson, B. A.

doi:10.1090/S0002-9939-1971-0281141-0

Families of mutually complementary topologies
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Proc. Amer. Math. Soc. 29 (1971), 362-368 Request permission

Abstract:

Several lattices of topologies on an infinite set are considered and bounds are given for the sup of the set of cardinals d such that there is a family of d mutually complementary topologies. Large classes of ${\aleph _0}$-topologies are shown to have ${\aleph _0}$-complements, and an example is given to prove that complementation is not, in general, a very selective topological operation.

References

B. A. Anderson and D. G. Stewart, $T_{1}$-complements of $T_{1}$ topologies, Proc. Amer. Math. Soc. 23 (1969), 77–81. MR 244927, DOI 10.1090/S0002-9939-1969-0244927-5
B. A. Anderson, A class of topologies with $T_{1}$-complements, Fund. Math. 69 (1970), 267–277. MR 281140, DOI 10.4064/fm-69-3-267-277
Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
S. P. Franklin, Spaces in which sequences suffice. II, Fund. Math. 61 (1967), 51–56. MR 222832, DOI 10.4064/fm-61-1-51-56
S. W. Williams and S. P. Franklin, Problems and Solutions: Solutions of Advanced Problems: 5468, Amer. Math. Monthly 75 (1968), no. 2, 208. MR 1534745, DOI 10.2307/2315929
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199

A sublattice of the lattice of topologies

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Paul S. Schnare, Infinite complementation in the lattice of topologies, Fund. Math. 64 (1969), 249–255. MR 242110, DOI 10.4064/fm-64-2-249-255
A. K. Steiner, The lattice of topologies: Structure and complementation, Trans. Amer. Math. Soc. 122 (1966), 379–398. MR 190893, DOI 10.1090/S0002-9947-1966-0190893-2
A. K. Steiner, Complementation in the lattice of $T_{1}$-topologies, Proc. Amer. Math. Soc. 17 (1966), 884–886. MR 193033, DOI 10.1090/S0002-9939-1966-0193033-4
E. F. Steiner and A. K. Steiner, Topologies with $T_{1}$-complements, Fund. Math. 61 (1967), 23–28. MR 230277, DOI 10.4064/fm-61-1-23-28
A. C. M. van Rooij, The lattice of all topologies is complemented, Canadian J. Math. 20 (1968), 805–807. MR 227920, DOI 10.4153/CJM-1968-079-9