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- W2015710740 abstract "W. J. Thron proved in 1962 that regularity and normality are lattice-invariant properties but not To and T1. In the present paper it is proved that complete regularity, compactness, local compactness, Lindel6f, second countability and connectedness are lattice-invariant properties. It is also proved that Hausdorff, complete normality, separability, and first countability are not lattice-invariant properties. 1. Two topological spaces are said to be lattice-equivalent if their lattices of open sets are isomorphic. It follows by duality that this is the case iff their lattices of closed sets are isomorphic. A topological property P is said to be lattice-invariant if a topological space latticeequivalent to a topological space having P has also P. W. J. Thron, in his paper [1], treated among others the lattice characterization of some separation axioms for topological spaces. He proved that To and T1 are not lattice-invariant. He also proved that regularity and normality are lattice-invariant. In this paper, we shall prove that complete regularity, compactness, local compactness (in a sense), Lindelof, second countability, and connectedness are lattice-invariant properties. We also prove that Hausdorff, complete normality, separability, and first countability are not lattice-invariant properties. 2. Let X be a topological space and ? the lattice of all open sets in X. The following results are evident. (i) X is compact iff for any family C 2 with V{D:D =X, there is a finite subfamily Do of D such that V{D:DE Secondary 0630, 5423, 5440, 5452, 5455." @default.
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- W2015710740 modified "2023-10-14" @default.
- W2015710740 title "Lattice-invariant properties of topological spaces" @default.
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- W2015710740 doi "https://doi.org/10.1090/s0002-9939-1970-0261549-9" @default.
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