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\(T_0\)-closure operators and pre-orders. (English) Zbl 1432.54017

Summary: It is well known that the lattice of closed subsets of any topological space is isomorphic to that of a \(T_0\)-topological space. This result is extended to lattices of closed subsets with respect to arbitrary closure operator on a set. Also, we establish a one-to-one correspondence between closure operators which are both algebraic and topological on a given set \(X\) and pre-orders on \(X\) and prove that this correspondence induces a one-to-one correspondence between topological algebraic \(T_0\)-closure operators on \(X\) and partial orders on \(X\).

MSC:

54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
06B30 Topological lattices
54A05 Topological spaces and generalizations (closure spaces, etc.)
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References:

[1] G. Birkhoff, Lattice Theory, Am. Math. Soc. Colloq. Publ. (AMS, Providence, USA, 1967), Vol. 25. · Zbl 0126.03801
[2] G. Gratzer, General Lattice Theory (Academic, New York, San Fransisco, 1978). · Zbl 0385.06014 · doi:10.1007/978-3-0348-7633-9
[3] G. F. Simmons, Introduction to Topology and Modern Analysis (McGraw-Hill, New York, 1963). · Zbl 0105.30603
[4] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra (Springer, New York, 1980). · Zbl 0478.08001
[5] U. M. Swamy, G. C. Rao, R. S. Rao, and K. R. Rao, “The lattice of closed subsets of a topological space,” South East Asian Bull. Math. 21, 91-94 (1997). · Zbl 0878.06004
[6] U. M. Swamy and R. S. Rao, “Algebraic Topological Closure Operators,” South East Asian Bull. Math. 26, 669-678 (2002). · Zbl 1043.06004 · doi:10.1007/s100120200071
[7] B. Venkateswarlu, R. Vasu Babu, and Getnet Alemu, “Morphisms on Closure spaces andMoore spaces,” Int. J. Pure Appl. Math. 91, 197-207 (2014). · Zbl 1303.06005 · doi:10.12732/ijpam.v91i2.6
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