definitionS of a topological space (reference)

Question

I know of at least 6 different ways to define a topological space:

1. with open sets,
2. with closed sets,
3. with nets,
4. with neighbourhoods,
5. with Kuratowski's closure operator,
6. and his interior operator.

I have a vague idea of how to work with the non-standard ones, and I am not sure about all equivalences, I do not have much reference. The last book where I looked for the kuratowski operators left me unsatisfied. (Moreover he told that it is possible to define a topology using a boundary operator, overlooking difficulties.)

I am asking for all the possible definitions of a topological space, or at least a reference to where to find them. Thank you.

Answer 1

Willard's General Topology has equivalences of the 6 that you listed. Some books also include an equivalence based on the frontier [= boundary] operator, but I'm not home now where all my books are. If you have access to a university library, then browse through the general topology texts on the shelves. The following two papers are among a couple I know of right now (I could probably dig up more if I was at home where all my stuff is) that include equivalences not in Willard's book.

José Ribeiro de Albuquerque (1910-1991), La notion de «frontière» en topologie [The notion of «frontier» in topology], Portugaliae Mathematica 2 #1 (1941), 280-289.

Miron Zarycki (1899-1961), Quelques notions fondamentales de l'analysis situs au point de vue de l'algèbre de la logique [Some fundamental notions of topology from the point of view of the algebra of logic], Fundamenta Mathematicae 9 (1927), 3-15.

(translation of 3 sentences near the beginning of the paper) In the present Note I consider some analogous systems of axioms for some other fundamental notions of topology, namely for the notions of exterior, of interior, of frontier and of border. I prove the equivalence of these systems to that of Mr. Kuratowski and I deduce some theorems concerning the fundamental properties of the mentioned notions. I wish to thank Mr. Kuratowski for his valuable advice concerning the final editing of this article.

(ADDED NEXT DAY)

This morning, while at home where all my math stuff is, I looked for some more references and found the following. I didn't bother with references for characterizations in terms of the interior operator (or nets, or neighborhoods, etc.) because these are quite common and in a lot of topology texts.

Papers

Alexander Abian (1923-1999), The derived set axioms for topology, Mathematica (Cluj) 12(35) #2 (1970), 213-215.

Abian shows that a topology for a set X can be characterized by a function $D:P(X) \rightarrow P(X)$ that simultaneously satisfies all of the following: (1) $D(\emptyset) = \emptyset;$ (2) For each $A,B \in P(X)$ we have $D(A \cup B) = D(A) \cup D(B);$ (3) For each $A \in P(X)$ we have: $D(A \cup D(A)) \subseteq A \cup D(A);$ (4) For each $x \in X$ we have $x \notin D(\{x\}).$

Shair Ahmad (1934- ), On the derived set operator (conference abstract #2), American Mathematical Monthly 71 #8 (October 1964), 956.

Abstract of a talk given at the annual spring meeting of the Minnesota Section of the Mathematical Association of America, College of St. Thomas (St. Paul, Minnesota), 9 May 1964: The four axioms for a derived set operator as given by [Frank Reese] Harvey are shown to be equivalent to three somewhat simpler axioms. A slight modification of these axioms renders them absolutely independent.

Kenneth Albert Henry Gravett (??-1966), A characterization of frontier, Proceedings of the Cambridge Philosophical Society 52 #1 (January 1956), 152-153.

Frank Reese Harvey (1941- ), The derived set operator, American Mathematical Monthly 70 #10 (December 1963), 1085-1086.

A topology for a set X can be characterized by a function $D:P(X) \rightarrow P(x)$ that simultaneously satisfies all of the following: (1) $D(\emptyset) = \emptyset;$ (2) For each $A,B \in P(X)$ we have $D(A \cup B) = D(A) \cup D(B);$ (3) For each $A \in P(X)$ we have: $x \in D(A)$ if and only if $x \in D(A - \{x\});$ (4) For each $A \in P(X)$ we have $D(A \cup D(A)) \subseteq A \cup D(A).$

Denis Arthur Higgs (1932-2011), Iterating the derived set function, American Mathematical Monthly 90 #10 (December 1983), 693-697.

A characterization for the topology on a set in terms of the derived set operator is given on p. 694.

Books

Hellen Frances Cullen (1919-2007), Introduction to General Topology, D. C. Heath and Company, 1968, xii + 427 pages.

Several alternative characterizations of a topological space are given in the subsection Extended and Conventional Definitions of a Topological Space on pp. 22-25, including the derived set operator. [Note: Her term "cotopology" refers to the collection of closed sets in a topological space.]

James Dugundji (1919-1985), Topology, Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon, 1966, xvi + 447 pages.

A characterization of the topology on a set in terms of the derived set operator is given on p. 73.

Michael Caesar Gemignani (1938 - ), Elementary Topology, Addison-Wesley Publishing Company, 1967, xi + 258 pages. [The 2nd edition was published by Addison-Wesley Publishing Company in 1972 (xi + 270 pages), and the 2nd edition was reprinted by Dover Publications in 1990.]

Exercise #5 on p. 59 (1972 2nd edition): Try to find a method for specifying a topology on a set $X$ by specifying Fr $A$ for each* $A \subset X.$ Do likewise for Ext. [The frontier (Fr) of a set $A$ is defined to be the intersection of the closure of $A$ and the closure of $X-A.$ The exterior (Ext) of $A$ is defined to be the complement of the closure of $A.$] There are no answers or hints for the exercises, and no references are given for this particular exercise.

Wolfgang Joseph Thron (1918-2001), Topological Structures, Holt, Rinehart and Winston, 1966, xii + 240 pages.

A characterization of the topology on a set in terms of the derived set operator is given on p. 53.

Ramaswamy Vaidyanathaswamy (1894-1960), Set Topology, 2nd edition, Chelsea Publishing Company, 1960, viii + 305 pages. [Reprinted by Dover Publications in 1999.]

On p. 58 (Example 13) three properties of the boundary operator are stated (hint provided) to characterize a topology on a set, where the boundary of $A$ is defined to be the set of all points in $A$ that do not belong to the interior of $A.$ On p. 58 (Example 15) four properties of the frontier operator are stated to characterize a topology on a set, where the frontier of $A$ is defined to be the union of the boundary of $A$ and the boundary of $A'$ $(A'$ is the derived set of $A).$ On p. 58 (Example 16) four properties of a certain $2$-variable frontier operator $F:P(X) \times P(X) \rightarrow P(X),$ defined by $F(A,B) = (A \cap \overline{B}) \cup (\overline{A} \cap B),$ are stated to characterize a topology on a set. On p. 59 (Example 20) four properties of the exterior operator are stated to characterize a topology on a set, where the exterior of $A$ is defined to be the interior of $X-A.$ Incidentally, all three of these characterizations can be found on pp. 58-59 of the first edition of Vaidyanathaswamy's book [Treatise on Set Topology. Part I, Indian Mathematical Society, Madris, 1947, vi + 306 pages], but I did not see any references to relevant literature in either the 1947 edition or the 1960 edition.

Student Theses

Mu-Lo Wang, Relations Among Basic Concepts in Topology, Master of Science thesis (under Henry Arnold Bradford), Oregon State University, June 1968, iv + 52 pages.

Author’s Abstract: It is well-known that a topology for a space can be described in terms of neighborhood systems, closed sets, closure operator or convergence as well as open sets. In fact, it is also possible to describe a topology in terms of interior operator or boundary operator. This paper is devoted to these descriptions and the interrelations between these basic topological notions.