Neighborhoods (Updated)

The past couple of posts on continuity and homeomorphism actually glossed over one really important point. I'm actually surprised no one called me on it; either you guys have learned to trust me, or else no one is reading this.

What I skimmed past is what a *neighborhood* is. The intuition for a
neighborhood is based on metric spaces: in a metric space, the neighborhood of a
point p is the points that are *close to* p, where close to is defined in terms of the distance metric. But not all topological spaces are metric spaces. So what's a neighborhood in a non-metric topological space?

A topological space is really nothing but a set of *points* with some structure of open and closed sets. The *neighborhood* of a point in a topological space is defined by the structure of that space.

The easiest way to define a neighborhood is in terms of the open and closed sets that define the topology. For a topological space (**T**, τ), a neighborhood of a point *x* ∈ **T** is either:

* An open set in τ containing *x*; or
* A superset of a neighborhood of *x* that is still a subset of **T**.

A set *N* that is a neighborhood of a point *x* is called an *open neighborhood* if it's a neighborhood of *every* point in *N*.

A more formal way of saying the above is to use a set of axioms. This particular formulation is based on the one in [Mendelson's][mendelson-book] text:

Given a topological space (**T**,τ):

1. For every point *x* ∈ **T**: there exists *at least* one neighborhood in (**T**,τ) containing *x*.
2. For every point *x* ∈ **T**, every neighborhood *N(x)* of *x* in (**T**, τ) contains *x*.
3. For every point *x* ∈ **T**, if *N(x)* is a neighborhood of *x* in (**T**,τ), and *M* is a superset of *N(x)*, then *M* is a neighborhood of *x*.
4. For any point *x* ∈ **T**, for any two neighborhoods *N(x)* and *M(x)* in (**T**,τ), *N(x) ∩ M(x)* is a neighborhood of *x*.
5. For any neighborhood *N(x)* of any point *x* in (**T**,τ), there is a neighborhood *M ⊂ N(x)* (*M* is a *proper* subset of N(x)) such that *N(x)* is a neighborhood of *every* point in *M*.

*(Note: I originally switched "M" and "N" at the end of axiom 5 above. I also decided to add a diagram to try to clarify what it means.)*

The last axiom is a bit hard to understand written out like that, but a diagram can do do wonders:

One neat thing about neighborhoods is that you can define topology entirely in terms of them. You can start with the neighborhoods, and then use the neighborhoods to define the open and closed sets, which in turn define the topology:

Take a set, *X*. A *neighborhood system* on *X* is an assignment of a *filter* *f*(x) to each point x ∈ *X*, such that:

1. ∀ x ∈ *X*, ∀ S ∈ F(x): x ∈ S.
2. ∀ x ∈ *X*, ∀ S ∈ F(x), ∃ R ⊂ S, ∀ r ∈ R, S ∈ F(r).

That second one looks awful, but it's not as bad as it looks. What is says is: for every neighborhood S of a point *x*, there is a subset R of that neighborhood such that S is a neighborhood of every point in R. In other words, it's equivalent to axiom 5 about neighborhoods above.

Given a neighborhood system, the open sets are the open neighborhoods.

One last little definition, since it's an important one, and it's defined in terms of neighborhoods. There's a very important group of topological spaces called *Hausdorff spaces*; they're also sometimes called *separated spaces*.

A topological space (**T**,τ) is a Hausdorff space if for any two distinct points in **T**, there are non-intersecting neighborhoods containing each of the points. Written formally: for every pair of points *x* and *y* ∈ **T**, there exists at least one pair of neighborhoods N(*x*) and M(*y*) where N(*x*) ∩ M(*y*) = ∅. So a Hausdorff space is a space where you can *separate* distinct points using neighborhoods. We'll see more about Hausdorff spaces later; and they come up all the time in mathematical analysis. Most intuitive topological spaces are Hausedorff.

There's a dreadful math joke that can help you remember what a Hausedorff space is. A space is Hausedorff is all of the points in it can be housed off from each other. (Argh!) Yeah, it's a really horrible joke. But I still remember it *more than 10 years* after I last studied topology; there's not many definitions I can remember without refreshing myself by browsing a textbook, but this one I remember. I suppose that says *something* about me, but I'd rather not know what.

[mendelson-book]: http://www.amazon.com/gp/redirect.html?link_code=ur2&tag=goodmathbadma-…