# definitionS of a topological space (reference)

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.

• I think you probably don't have to ask about the first four. Presumably equivalence of the Kuratowski operator definitions with any of the first four would be enough for you, wouldn't it?
– Danu
Apr 27, 2017 at 9:22
• You can also use filter convergence and even boundary-operators. Apr 27, 2017 at 11:37
• "all the possible definitions of a topological space" is likely an infinite set. Even if you restrict your attention to ones that have appeared in the literature somewhere or other (e.g. Ralph Kopperman's paper "All topologies come from generalized metrics") it will be a fairly large list. Apr 27, 2017 at 15:06

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.

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.

• I was reading Willard's book these past days, where for the first time I found the axioms for the topology through nets, and was pleased to find the characterizations I know presented together in one book, even though not with much explicit examples of extensions and usage. The french papers left me stunned for the "categorical" notation on the subsets of the space, but given time I think even without the language I will be able to read them. Given time I will look into all the papers, and as soon as I can I will accept the answer! :) THANK YOU. Apr 30, 2017 at 10:40
• Dave your reference books are really hard to find. But I made some good progress. Now I just need to understand and find a duality for the derived set. Thank you. May 11, 2017 at 8:08