The topology posts have been extremely abstract lately, and from some of the questions
I've received, I think it's a good idea to take a moment and step back, to recall just
what we're talking about. In particular, I keep saying "a topological space is just a set
with some structure" in one form or another, but I don't think I've adequately maintained
the *intuition* of what that means. The goal of today's post is to try to bring back
at least some of the intuition.
So let's recall just what a topological space is. Our definition from the [very beginning of
the topology series was:][top-space]
A topological space is a set **X**, and a collection **T** of subsets of **X** where the following conditions hold:
1. ∅ ∈ **T** and **X** ∈ **T** *(The empty set and the entire set **X** are both members of **T**.)
2. (∀ C ∈ **2****T**): (∀ c ∈ C : ∪(c ∈ C)c ∈ **T**) *(The union of any set of elements of **T** must be a set in **T**.)*
3. ∀ s,t ∈ **T**: s ∩ t ∈ **T** *(The intersection of any pair of members of **T** is also a member of **T**.)*
**T** is the structure imposed on the set **X** that we've been talking about. But just what does that mean? It's really a very fancy way of taking the concept of *closeness* or *adjacency* and abstracting it out so the concept of *distance* isn't needed. In a topological space, we don't care whether we can measure *how far* it is from a point *A* to a point *B*; but we *do* care
whether we can meaningfully ask "Is B closer to A than C?" or "Is A adjacent to B?". The
structure of the open subsets in a topological space gives us a way of talking about that.
How can we answer those questions? By playing with neighborhoods - that is, expanding "shells" of points around a particular given point. Suppose we want to ask "**Is *B* closer to *A* than it is to *C*?**"
Take a sequence *S* of expanding subsets around *B* something like the open balls in a metric space - that is, a sequence of subsets that are uniformly growing larger, but always including all of the points in the subsets that precede them. If *A* becomes an element of the sets in the sequence *before* *C* does, then *with respect to* the sequence *S*, *A* is closer to *B* than *C* is. In a topological space, you *cannot* in general define something like *closer to* in a universal way; there are many ways that the open sets can be constructed, and it's entirely possible to have *many different* ways of describing closeness based on different
constructions, and there's no reason to prefer one of them over the other.
That basic concept - what points are *next to* what points, and what points are *close to* what other points - is what's defined by the open-set structure of the topological space. The notion of "close to" is based completely on subset inclusion relationships; you can't necessarily assign a number to the distance between two points (if you could, we'd call it a metric space!), but you can always look at the subset inclusion relationships to understand where the points lie in relation to each other.
To bring this forward a bit, in my messed up post about sheaves, one of the key ideas was
the gluing axiom. The gluing axiom says, very basically, that you can map between *sets* in an overlap between two sections; it does *not* do a coordinate transformation. That misunderstanding was caused by a combination of some genuinely subtle distinctions, and some
dreadful sloppiness on my part.
When we talked about gluing manifolds, what we were doing is forming manifolds by mapping
sections of *euclidean spaces* onto manifolds, and gluing them together with coordinate
transformations. That *is* gluing, and the theoretical basis for it *is* sheaves and
the gluing axiom that allows them to be combined. But the important distinction is that what we were gluing was sections of euclidean spaces - and euclidean spaces have a standard metric, and we describe the glue maps in terms of that standard metric.