Talk:Axiomatic foundations of topological spaces

Definition problem (resolved)

The "Definition via convergent filters" yields pretopological spaces, which - as their name indicates - are not (yet) topological spaces. Topological spaces may be defined by way of convergence of filters, but this needs a more involved axiom. The relation with topological spaces mentioned in this section yields a reflector, not an isomorphism.

The "Isotonicity" condition for an operator P(X) → P(X) was used once for the first axiom involving closure operators, and once for interior operators (thus, with oppposite meanings)! Furthermore, in A Course in Universal Algebra, or [1], "isotone" is used to mean monotone which is the correct use if we compare it to Wikipedia's definition of antitone. Unfortunately, none of the concepts described here as isotone means monotone. This has been corrected.

Add exterior, boundary, derived set, and co-derived set operators.

The page on Kuratowski closure operators, mentions interior, exterior, and boundary operators, but only gives axioms for the interior operators, which are also on this page (https://en.wikipedia.org/wiki/Kuratowski_closure_axioms#Interior,_exterior_and_boundary_operators). This paper (https://doi.org/10.1016/j.entcs.2019.07.016) includes axioms for these operators as well as axioms for derived set, and co-derived set operators and gives proofs for their equivalence. These other operator axioms should be added for completeness. I suggest the page uses the standard axioms for the boundary operator and the alternative axioms for the derived set operator. 129.2.192.184 (talk) 17:54, 7 January 2024 (UTC)[reply]

Additional definition by "contact relation"

I cannot read German but idea comes from G. Aumann and was intended as a more intuitive definition for learning about closed sets as sets that a point is in "contact" with.

Explained in Contact Relations with Applications by Gunther Schmidt and Rudolf Berghammer

p4 of 16 defines:

3 Contact Relations If we formulate Aumann’s original definition of a contact relation given in [1] in our notation, then a relation A : X ↔ 2X is an (Aumann) contact relation if the following conditions hold.

(A1 ) ∀ x : Ax,{x}

(A2 ) ∀ x, Y, Z : Ax,Y ∧ Y ⊆ Z → Ax,Z

(A3 ) ∀ x, Y, Z : Ax,Y ∧ (∀ y : y ∈ Y → Ay,Z ) → Ax,Z

I would summarize as a relation A from points to subsets is a "contact relation" if:

1. Each point is in contact with the subset consisting of itself.

2. Each point is in contact with any superset of any subset it is in contact with.

3. Each point is in contact with any subset that all the members of any subset it is in contact with are all in contact with.

Related to formal concept lattices and Theory of Convex Structures by M.L.J. van de Vel:

Chapter 1: Abstract Convex Structures, p44, pdf 60 of 556. Copied from scanned pdf and pasted below - sorry I don't have time to format properly but I am not competent to work on writing it up for publication and anyone potentially interested in doing so can easily follow the links provided above.

My impression is that the definition below first defines closed sets and extends it to abstract convexity (which is just an intersection closure space plus upwards chain closure). But the original idea by Aumann was for teaching definitions in topology.

Further Topics 2.20. Convexity and social affinity (Aumann [1971]). Let S (for “society”) and I (for “interests”) be sets, and let i : S -+ 2’ be a function such that i ( x ) # 0 for all x E S. A person p has social finit)‘ with a finite set of individuals T E S provided his personal interests i ( p ) are shared by the members of T: i ( p ) G i ( T ) = U { i ( x )I X E T } . Show that this prescription satisfies the axioms of betweenness. This leads to a convexity consisting of all sets C E S which are stable in the sense that p E C whenever p has social affinity with a finite set of persons in C. Conversely, show that every convex structure arises from an “interest” function as described above. Hint: be interested in concave sets. 2001:8003:4B08:5C01:DFCA:7611:D4A4:FE2C (talk) 02:42, 28 April 2024 (UTC)[reply]