# 8. Selected References

The references below are mostly restricted to those that specifically involve, entirely or partly, various generalized closure operator notions. Thus, the literature on generalized topological notions dealing with the vast zoo of various semi-open set notions, various types of separation axioms in generalized spaces, various generalized continuity notions, category-theoretic connections, etc. are not included unless I thought it had specific relevance to something discussed here.

**[1]** Peter Henry George Aczel, *An introduction to inductive definitions*, pp. 739-782 in Jon Kenneth Barwise (editor), **Handbook of Mathematical Logic**, North-Holland, 1977, xi + 1165 pages.

**[2]** Jon Kenneth Barwise, **Admissible Sets and Structures. An Approach to Definability Theory**, Perspectives in Mathematical Logic #13, Springer-Verlag, 1975, xiii + 394 pages.

**[3]** Jon Kenneth Barwise and Lawrence Stuart Moss, **Vicious Circles. On the Mathematics of Non-Wellfounded Phenomena**, CSLI Lecture Notes #60, 1996, x + 390 pages.

**[4]** Mary Katherine Bennett, *Convexity closure operators*, **Algebra Universalis** 10 #3 (1980), 345-354.

**[5]** Garrett Birkhoff, **Lattice Theory**, corrected reprint of 3rd edition, American Mathematical Society Colloquium Publications #25, American Mathematical Society, 1979, vi + 418 pages.

**[6]** Jürgen Bliedtner and Peter Albert Loeb, *The optimal differentiation basis and liftings of* $L^{\infty}$, **Transactions of the American Mathematical Society** 352 #10 (October 2000), 4693-4710.

**[7]** Donald Jerome Brown and Roman Suszko, *Abstract Logics*, **Dissertationes Mathematicae** 102 (1973), 9-41.

**[8]** Stanley Neal Burris, **Theory of Pre-Closures**, Ph.D. dissertation (under Allen Seymour Davis), University of Oklahoma, 1968, iv + 47 pages.

This is a study of closure operators that satisfy (semi-K1) and (K3).

**[9]** Eduard Čech, **Topological Spaces**, revised and translated by Zdeněk Frolík and Miroslav Katětov, John Wiley and Sons, 1966, 893 pages.

Čech published a survey of his Čech closure operator work in 1937: *Topologické prostory*, **Časopis pro Pěstování Matematiky a Fysiky** 66 #4 (1937), D225-D264.

**[10]** Ákos Császár, *Generalized open sets*, **Acta Mathematica Hungarica** 75 #1-2 (April 1997), 65-87.

In this paper and the next paper, Császár studies various topological ideas in the context of a generalized topological space defined by a monotone closure operator, although for many of the results he (explicitly) assumes additional hypotheses, as needed.

**[11]** Ákos Császár, *On the* $\gamma$-interior and* $\gamma$-closure of a set*, **Acta Mathematica Hungarica** 80 #1-2 (July 1998), 89-93.

**[12]** Sterling Gene Crossley, **Semi-Topological Properties and Related Topics**, Ph.D. dissertation (under Shelby Keith Hildebrand), Texas Technological College [*after 1968*: Texas Tech University], 1968, iii + 76 pages.

**[13]** Dirk van Dalen, Hans Cornelis Doets, and Henricus Cornelius Maria de Swart, **Sets: Naïve, Axiomatic and Applied**, International Series in Pure and Applied Mathematics #106, Pergamon Press, 1978, xviii + 339 pages.

**[14]** Mahlon Marsh Day, *Convergence, closure and neighborhoods*, **Duke Mathematical Journal** 11 #1 (March 1944), 181-199.

**[15]** Vladimir Devidé, *On monotone mapings* [sic] *of the power-set*, **Portugaliae Mathematica** 21 #2 (1962), 111-112.

**[16]** Keith James Devlin, **The Joy of Sets. Fundamentals of Contemporary Set Theory**, 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, 1993, x + 192 pages.

**[17]** Szymon Dolecki and Gabriele H. Greco, *Topologically maximal pretopologies*, **Studia Mathematica** 77 #3 (1984), 265-281.

**[18]** Herbert Bruce Enderton, **Elements of Set Theory**, Academic Press, 1977, xiv + 279 pages.

**[19]** Preston Clarence Hammer, *General topology, symmetry, and convexity*, **Transactions of the Wisconsin Academy of Sciences, Arts and Letters** 44 (1955), 221-255.

**[20]** Preston Clarence Hammer, *Kuratowski’s closure theorem*, **Nieuw Archief voor Wiskunde** (3) 8 (1960), 74-80.

**[21]** Preston Clarence Hammer, *Extended topology: reduction of limit functions*, **Nieuw Archief voor Wiskunde** (3) 9 #1 (1961), 16-24.

**[22]** Preston Clarence Hammer, *Semispaces and the topology of convexity*, pp. 305-316 in Victor LaRue Klee (editor), **Convexity**, Proceedings of Symposia in Pure Mathematics 7, American Mathematical Society, 1963.

**[23]** Preston Clarence Hammer, *Extended topology: set-valued set-functions*, **Nieuw Archief voor Wiskunde** (3) 10 #? (1962), 55-77.

**[24]** Preston Clarence Hammer, *Extended topology: additive and subadditive subfunctions of a function*, **Rendiconti del Circolo Matematico di Palermo** (2) 11 #3 (1962), 262-270.

**[25]** Preston Clarence Hammer, *Extended topology: perfect sets*, **Portugaliae Mathematica** 23 #1 (1964), 27-34.

**[26]** Preston Clarence Hammer, *Extended topology: the continuity concept*, **Mathematics Magazine** 36 #2 (March 1963), 101-105.

**[27]** Preston Clarence Hammer, *Extended topology: Continuity I*, **Portugaliae Mathematica** 23 #2 (1964), 77-93.

**[28]** Preston Clarence Hammer, *Extended topology: connected sets and Wallace separations*, **Portugaliae Mathematica** 22 #4 (1963), 167-187.

**[29]** Preston Clarence Hammer, *Extended topology: structure of isotonic functions*, **Journal für die reine und angewandte Mathematik** 213 #3-4 (1964), 174-186.

**[30]** Felix Hausdorff, *Gestufte räume*, **Fundamenta Mathematicae** 25 (1935), 486-502.

**[31]** Arie Hinkis, **Proofs of the Cantor-Bernstein Theorem. A Mathematical Excursion**, Science Networks / Historical Studies #45, Birkhäuser, 2013, xxiv + 429 pages.

**[32]** Peter Greayer Hinman, **Recursion-Theoretic Hierarchies**, Perspectives in Mathematical Logic #7, Springer-Verlag, 1978, xii + 480 pages.

**[33]** Karel Hrbacek and Thomas J. Jech, **Introduction to Set Theory**, 3rd edition, Pure and Applied Mathematics #220, Marcel Dekker, 1999, xii + 291 pages.

**[34]** Robert Edward Jamison, **A General Theory of Convexity**, Ph.D. dissertation (under Victor LaRue Klee), University of Washington, 1974, v + 120 pages.

**[35]** Varadachariar Kannan, **Ordinal Invariants in Topology**, Memoirs of the American Mathematical Society 32 #245 (July 1981), vi + 164 pages.

**[36]** T. Kavitha, **Some Problems on Čech Closure Spaces**, Ph.D. dissertation (under P. T. Ramachandran), University of Calicut (Kerala, India), August 2017, viii + 139 pages.

*Generalized Čech closure operators* $-$ closure operators that satisfy (semi-K1), (K2), (K3) $-$ are studied in Sections 5.4-5.4 on pp. 120-131.

**[37]** Darrell Conley Kent and Gary Douglas Richardson, *The decomposition series of a convergence space*, **Czechoslovak Mathematical Journal** 23 #3 (1973), 437-446.

**[38]** Bronisaw Knaster, *Un théorème sur les fonctions d’ensembles* [A theorem on functions of sets], **Annales de la Société Polonaise de Mathématique** [= Rocznik Polskiego Towarzystwa Matematycznego] 6 (1927), 133-134.

This is not a published paper, but rather an abstract from the section titled *Comptes-rendus des séances de la Société Polonaise de Mathématique Section de Varsovie* [Reports of the sessions of the Warsaw Section of the Polish Mathematics Society] (session taking place 9 December 1927). The results were obtained jointly with Tarski and presented by Knaster.

**[39]** William B. Koenen, *The Kuratowski closure problem in the topology of convexity*, **American Mathematical Monthly** 73 #7 (August-September 1966), 704-708.

**[40]** Karel Koutský, *Čech's topological seminar in Brno, 1936-1939*, **Časopis pro Pěstování Matematiky** 90 #1 (1965), 104-117.

**[41]** Yinbin Lei and Jun Zhang, *Generalizing topological set operators*, **Electronic Notes in Theoretical Computer Science** 345 (28 August 2019), 63-76.

**[42]** Jaroslav Lukeš, Jan Malý, and Luděk Zajíček, **Fine Topology Methods in Real Analysis and Potential Theory**, Lecture Notes in Mathematics #1189, Springer-Verlag, 1986, x + 472 pages.

**[43]** Jacek Malinowski, *On generalizations of consequence operation*, **Bulletin of the Section of Logic** (University of Łódź) 31 #2 (2002), 135-143.

**[44]** Zlatko P. Mamuzić, **Introduction to General Topology**, translated by William Joseph Pervin and James Leo Sieber and Robert C. Moore, P. Noordhoff (Netherlands), 1963, 159 pages.

**[45]** Norman Marshall Martin and Stephen Randall Pollard, **Closure Spaces and Logic**, Mathematics and Its Applications #369, Kluwer Academic Publishers, 1996, xviii + 230 pages.

The closure spaces of this book satisfy (semi-K1), (K3), and $T^2 \leq T.$ That is, from the topological closure operator axioms we drop half of (K1), (K2), and half of (K4).

**[46]** Elliott Mendelson, **Schaums Outline of Theory and Problems of Boolean Algebra and Switching Circuits**, Schaums Outline Series, McGraw-Hill Book Company, 1970, viii + 213 pages.

**[47]** Eliakim Hastings Moore, *Introduction to a Form of General Analysis*, pp. vii-viii + 11-50 in American Mathematical Society Colloquium Publications #2, Yale University Press, 1910, x + 222 pages.

Moore’s *extensionally attainable* property (defined on middle of p. 59) is often cited as an early, and perhaps the earliest, formulation of a general closure operator notion. Supposedly it is equivalent to a closure operator $T$ on a set that satisfies (semi-K1), (K3), and $T^2 \leq T$ (e.g. pp. 16-17 of Martin/Pollard **[45]**), but I have not attempted to verify this or consider whether such claims might be somewhat ahistorical. For a historical study of Moore’s General Analysis program, see Reinhard Siegmund-Schultze, *Eliakim Hastings Moore’s “General Analysis”*, **Archive for History of Exact Sciences** 52 #1 (January 1998), 51-89.

**[48]** Yiannis Nicholas Moschovakis, **Elementary Induction on Abstract Structures**, Studies in Logic and the Foundations of Mathematics #77, North-Holland, 1974, x + 218 pages.

**[49]** Yiannis Nicholas Moschovakis, **Descriptive Set Theory**, Studies in Logic and the Foundations of Mathematics #100, North-Holland, 1980, xii + 637 pages.

**[50]** Yiannis Nicholas Moschovakis, **Notes on Set Theory**, 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, 2006, xii + 276 pages.

**[51]** Josef Novák, *On some problems concerning multivalued convergences*, **Czechoslovak Mathematical Journal** 14 #4 (1964), 548-561.

**[52]** Horst Wolfram Pohlers, **Proof Theory**, Lecture Notes in Mathematics #1407, Springer-Verlag, 1989, viii + 213 pages.

**[53]** Marlon Cecil Rayburn, *On the lattice of* $\sigma$-*algebras*, **Canadian Journal of Mathematics** 21 (1969), 755-761.

**[54]** Sheldon Theodore Rio, **On the Hammer Topological System**, Ph.D. dissertation (under Henry Arnold Bradford), Oregon State College [*after 1961*: Oregon State University], June 1959, iv + 53 pages.

**[55]** David Niel Roth, **Čech Closure Spaces**, Master of Arts thesis (under John Warnock Carlson), Emporia State University (Emporia, Kansas), July 1979, iv + 25 pages.

**[56]** David Niel Roth and John Warnock Carlson, *Čech closure spaces*, **Kyungpook Mathematical Journal** 20 #1 (June 1980), 11-30.

**[57]** Marcin Jan Schroeder, *Modification of pre-closure spaces as closure/interior operations on the lattice of pre-closure operators*, **RIMS Kôkyûroku** #1712 (Algebras, Languages, Algorithms in Algebraic Systems and Computations), Research Institute for Mathematical Sciences (Kyoto University, Japan), September 2010, 148-155.

**[58]** Gideon Ernst Schwarz, *A note on transfinite iteration*, **Journal of Symbolic Logic** 21 #3 (September 1956), 265-266.

See the half-page review by Dana Stewart Scott in **Journal of Symbolic Logic** 22 #3 (September 1957), p. 303.

**[59]** Wacław Franciszek Sierpiński, **Algèbra des Ensembles** [Algebra of Sets], Monograe Matematyczne #23, Nakadem Polskiego Towarzystwa Matematycznego (Warszawa-Wrocaw), 1951, ii + 205 pages.

**[60]** Wacław Franciszek Sierpiński, **General Topology**, translated by Cypra Cecilia Krieger, 2nd edition, Mathematical Expositions #7, University of Toronto Press, 1952, xii + 290 pages.

**[61]** Bärbel M. R. Stadler and Peter Florian Stadler, *Generalized topological spaces in evolutionary theory and combinatorial chemistry*, **Journal of Chemical Information and Computer Sciences** 42 #3 (May 2002), 577-585.

**[62]** Bärbel M. R. Stadler and Peter Florian Stadler, *Basic properties of closure spaces*, unpublished manuscript, 29 January 2002 (web release date), 20 pages.

**[63]** Alfred Tarski, *A lattice-theoretical fixpoint theorem and its applications*, **Pacific Journal of Mathematics** 5 #2 (June 1955), 285-309.

The main results were announced in *A fixpoint theorem for lattices and its applications* (abstract #496, received 27 June 1949), **Bulletin of the American Mathematical Society** 55 #11 (November 1949), 1051-1052.

**[64]** Wolfgang Joseph Thron, **Topological Structures**, Holt, Rinehart and Winston, 1966, x + 240 pages.

**[65]** Wolfgang Joseph Thron, *What results are valid on closure spaces*, **Topology Proceedings** 6 (1981), 135-158.

**[66]** Marcel Lodewijk Johanna van de Vel, **Theory of Convex Structures**, North-Holland Mathematical Library #50, Elsevier Science Publishers B.V., 1993, xv + 540 pages.

**[67]** Stephen W. Willard, **General Topology**, Addison-Wesley, 1970, xii + 369 pages.

**[68]** Ryszard Wójcicki, **Theory of Logical Calculi. Basic Theory of Consequence Operations**, Synthese Library: Studies in Epistemology, Logic, Methodology, and Philosophy of Science #199, Kluwer Academic Publishers, 1988, xviii + 473 pages.

**[69]** Rafał Ryszard Zduńczyk, *Simple systems and closure operators*, **Bulletin de la Société des Science et des Lettres de Łódź, Recherches sur les Déformations**, 66 # 3 (2016), 65-72.

Introduction to General Topologyby Zlatko P Mamuzić (1963; review on pp. 127-128 here) begins by considering "closure operators" satisfying NO axioms(continued)$\endgroup$Topological Spacesby Eduard Čech (1966), is a very complete study spaces in which idempotency is not assumed for the closure operation. See also this. $\endgroup$7more comments