Motivation of the definition of topology

[kellhus]

The author writes in Chapter 1, Section 12:

The definition of a topological space that is now standard was a long time in being formulated. Various mathematicians—Fréchet, Hausdorff, and others—proposed different definitions over a period of years.

What motivates the current definition of topology / open sets?

[kellhus]

Added a commentary on the definition — from Lee's Introduction to Topological Manifolds and from Brown's Topology and Groupoids.

I am not sure what Fréchet and Hausdorff did, and how their definitions were considered to be insufficient. Perhaps someone else could chime in?

[AddemF]

I'm not sure if I'm supposed to respond to your comment in another comment or if there's some way to attach my comment to your other comment ... but about the other definitions of a topology, I'm not sure what the others proposed as definitions and I'd be interested to find out. But I have at least a guess about Hausdorff: Some spaces are called Hausdorff spaces, and you'll see the definition of that in a later section/chapter. I wonder if the thing that we today call Hausdorff spaces are what he called a topology in general.

[pqnelson]

It has to do with the definition of continuous functions, really. I cannot get into detail at the moment (I'm at work), but think about a topology as a subset of the power set (satisfying various axioms).

Use the contravariant powerset functor to consider the possible subsets of your given set. Continuous functions are given "for free" in this method, under the contravariant powerset functor.

This is a very rough sketch, probably barely coherent at all, but the short answer: it's the only possible structure such that continuous functions "makes sense".

[pqnelson]

Here's my rambling thoughts...

Contravariant Power Set Functor

If you don't know what a "functor" is (or what "contravariant" means), don't worry: I won't use these terms ever again.

Now, how do we describe a subset $A$ of $X$? One clever way is with indicator functions. That is to say, we have a function $\chi{A}\colon X\to{0,1}$ such that $\chi{A}(a)=1$ if $a\in A$ and $\chi_{A}(x)=0$ otherwise.

Consequently, the power set of $X$ may be described as the set of all indicator functions defined on $X$.

Notation. I am just going to write $2$ for the set ${0,1}$, and I trust everyone will know what I mean by it.

Behaviour on Functions

If we have a function $f\colon X\to Y$, and we consider the power sets $\mathcal{P}(X)$ and $\mathcal{P}(Y)$, then can we "construct" a "natural" function using $f$ on the power set?

Well, if we describe the power sets as indicator functions, then $S\in\mathcal{P}(Y)$ means $S\colon Y\to{0,1}$...since that's the definition of an indicator function.

We have a "natural" operation (function composition) to consider: $S\circ f\colon X\to 2$. Right? Because $X\to Y\to 2$ is the composition going on here.

Observe then that $f\colon X\to Y$ gives us a natural mapping $\mathcal{P}(Y)\to\mathcal{P}(X)$ by sending $\chi{B}\mapsto\chi{B}\circ f$. It is no coincidence the natural mapping is "backwards".

Punchline

Well, in some sense we still have an indicator function...but the function $(\chi{B}\circ f)(x)=1$ if $x\mapsto f(x)\in B$, and $(\chi{B}\circ f)(x)=0$ if $x\mapsto f(x)\notin B$,

This means that $(\chi{B}\circ f)$ is the indicator function for the preimage of $f$ on $B$._

More generally, when we consider power sets in terms of indicator functions, we can promote functions on the underlying sets to mappings on the "power sets" but "in the opposite direction"...i.e. $f\colon X\to Y$ gives us a natural function $\mathcal{P}(Y)\to\mathcal{P}(X)$. This natural function simply gives us the preimage of the subset (determined by the indicator function) of the underlying mapping $f$.

What does this mean for topology?

Well, for a set $X$, a topology $\mathcal{T}(X)$ is a subset of the powerset $\mathcal{P}(X)$. The "natural mappings" for a topological space would be a restricted subcollection of the "natural mappings" for the powerset.

Now, how do we restrict this subcollection of mappings?

We already have a definition of continuity from calculus ("For each $\epsilon>0$ there is a corresponding $\delta>0$ such that blah blah blah"). If we attempted to translate it to topology via this "power set with indicator functions" approach, then we'd define a continuous function as such that the preimage of an open set is open.

Amazingly enough, that's the intuition captured by the $\epsilon-\delta$ definition.

...but what's an "open set"?

What We Demand For Continuity

Demand 0. Well, naively we want the identity function to be continuous. This necessarily means that $X\in\mathcal{T}(X)$.

Demand 1. Points should be continuous $1\to X$. But the preimage of an open set $M\subseteq X$ which is not in the image should be handled properly...this requires the empty set being open.

Desired Property. More generally, if $f\colon X\to Y$ is continuous at $x\in X$, then we expect for any neighborhood $N\in\mathcal{T}(Y)$ containing $f(x)\in N$ there is a corresponding neighborhood $M\in\mathcal{T}(X)$ such that $f(M)\subseteq N$. The function is just "continuous" if it is continuous everywhere on $X$. This is just the definition of continuous functions we want the topology to preserve.

Demand 2. If $N{1},N{2}\in\mathcal{T}(Y)$ contain $f(x)$, then $N{1}\cap N{2}$ clearly contains the point $f(x)$...and we'd like continuity there. Hence we would like there to be a corresponding $M'\in\mathcal{T}(X)$ such that $f(M')\subseteq N{1}\cap N{2}$.

(This would imply that the finite intersection of open sets is open.)

Demand 3. If we consider the family of sets ${N{j} : f(x)\in N{j}}$, then we would like continuity for the set $N=\bigcup{j} N{j}$. This implies the union of arbitrary open sets is open.

Closing Thoughts

The basic argument presented here also applies in Measure theory, to Sigma Algebras. We just need to use the notion of a measurable function to determine the axioms of the structure.

[AddemF]

Gotta say, that comment by pqnelson is what I find beautiful in Math--an elegant and insightful second perspective on a familiar object.

[pqnelson]

I'm not sure if everyone is aware of the "Stuff, Structure, and Properties" taxonomy for definitions, but if you know some, e.g., linear algebra...you'll know that a "vector space" consists of a set of vectors equipped with (i) scalar multiplication, (ii) vector addition, and (iii) a zero vector such that (a bunch of equations hold).

In fact, most definitions in math are like that...a gadget consists of stuff (usually a set or two or three) equipped with some structure (usually binary operators, sometimes unary operators like inversion, sometimes special elements like the zero-vector) such that some properties hold (usually a bunch of equations).

Topology is like this, but a little different. The "structure" is a collection of subsets...or from the way I presented it, a collection of indicator functions.

We have these other things...usually "functions of gadgets", or gadget-o-morphisms. They "preserve the structure"---what? Well, with vector spaces, they are linear maps $f\colon V\to W$. They are such that $f(x\vec{v}+y\vec{w})=xf(\vec{v})+yf(\vec{w})$ for scalars $x$ and $y$, and vectors $\vec{v},\vec{w}\in V$. Observe this preserves scalar multiplication, vector addition, and the zero-vector as identity element for vector addition. (Woah! They preserve the "structure"!)

But look: continuous functions preserve the "topological structure"! When I first realized this, it took me quite a while after learning the "stuff, structure, properties"..."paradigm". I just thought I'd like to share that.

(Also: it's really the morphisms that tell us all the interesting things. Think about it: when I contrived the axioms for a topology, it was from demanding continuous functions make sense. That's all we need.)