Reflexivity and transitivity of nearness in topology

Question

I'm reading Michael Henle's A Combinatorial Introduction to Topology where he defines topological spaces and nearness thusly:

A topological space is a set together with the choice of a class of subsets N of (each of which is called a neighborhood of its points) such that

(a) Every point is in some neighborhood.

(b) The intersection of any two neighborhoods of a point contains a neighborhood of that point.

[...]

Let be a topological space. Let A be a subset of and P be a point of . P is near A, written P ← A, if every neighborhood of P contains a point of A.

I'm more familiar with category theory than topology, so I tried to extend nearness to a binary relation between sets so I could use it to define morphisms between subsets of :

Given A, B ⊆ , say A ← B if ∀ P ∈ A, P ← B

But then after rereading the normal definition of nearness, I realized that there's no requirement that this binary relation is reflexive or transitive, so this definition is useless for defining a category.

For example, I could define a topological space on ℝ² where a neighborhoods of a point are open disks around the reflection of that point about the diagonal x=y; that is a neighborhood of (a,b) is { (x,y) | (x - b)² + (y - a)² < δ² }.

In this topological space, my extension on nearness is neither reflexive nor transitive. (a,b) ← { (b,a) }, and (b,a) ← { (a,b) }, but (a,b) ← { (a,b) } only if a = b.

My question is, how often do topological spaces like this come up? Is there a name for topological spaces where the extension of nearness to a binary relationship between sets is non-reflexive and/or non-transitive?

Answer 1

I talked about this at lunch with a topologist, and I think I was wrong.

I misread the definition - I missed the fact that every neighborhood is a neighborhood of each of its points. That is, for any topological space , a neighborhood A is a neighborhood of P if and only if P ∈ A.

In the usual plane topology, I'd been thinking of the open disc centered at a point as a neighborhood belonging to only the center point, not a neighborhood belonging to all of its points.

My example using ℝ² with neighborhoods reflected about x=y is then not a valid topological space.

You can extend nearness to a reflexive and transitive binary relation between subsets of .

Let A, B &subseteq; . Say A is near B if for all P ∈ A, P is near B.

• Reflexivity: ∀ P ∈ A, ∀ neighborhoods X of P, P ∈ X ⇒ ∃ some point of A in X, so P is near A.

  Therefore A is near A.

• Transitivity: Suppose ∃ A, B, C &subseteq; .

  Consider neighborhood X of P ∈ A.

  If A is near B, then P is near B, so ∃ Q ∈ B s.t. Q ∈ X. This means X is also a neighborhood of Q ∈ B.

  If B is near C, then Q is near C, so ∃ R ∈ C s.t. R ∈ X.

  Therefore P ← C, so A ← B and B ← C implies A ← C.