What is Topology?

For those who are familiar with real analysis and metric spaces, we all know that in a metric space, we can talk about sets being open and closed, we can define interior of a set, closure of a set, limit points and adherent points.

The Sphere, The Torus & The Genus-2-Orientable Surface

The aim of mathematicians, often is to go from particular to general. So, it is reasonable to ask the question:

“Can we define the aforementioned notions (open sets, closed sets, limit points etc.) without having a distance / metric defined on the space?”

The answer turns out to be a big ‘YES’, we do not really need a distance function to be able to define the concepts of open sets, closed sets, interior, closure, limit points etc. In fact, there are example in plenty, where we have a space equipped with all these notions but which are not ‘metrizable‘, i.e. we cannot give any compatible distance function to those spaces.

One way to go from special to general is the method of abstraction. We follow the same method to take a leap from metric spaces to general topological spaces.

First of all, we should recall some key properties of open sets (or closed sets) from metric spaces.

1. The empty set and the full set is open.

2. Union of any collection of open sets is again open.

3. Intersection of finitely many open sets is open.

We are now going to define what is called a ‘topology’ on a given non-empty set using the above properties as the defining axioms. This would allow us to talk about open sets in this general setting.

Suppose, $X$ is a non-empty set. Let $\tau$ be a collection of subsets of $X$ . We say, that $\tau$ is a topology on $X$ if it satisfies the three axioms:

$1.\space$\phi,X\in\tau$$
$2.\space$\{U_i:i\in I\}\subset\tau\implies\bigcup\limits_i U_i\in\tau$$
$3.\space$U_1,U_2\in\tau\implies U_1\cap U_2\in\tau$$

The set $X$ along with $\tau$ satisfying (1)-(3) is called a topological space and is written as a pair $(X,\tau)$ . The elements of $\tau$ are called the open sets of the topology $\tau$ .

So, let’s understand what we did. We took the essential properties of open sets from metric spaces and then we took those properties as the defining axioms of a topology on $X$ and we are now calling the elements of the topology $\tau$ as the open sets of the topological space $(X,\tau)$ .

Thus, in the back of mind, we had the intention to define $\tau$ in a way, so that it represents the collection of all open sets of the space $X$ and each of the elements of $\tau$ represents an open set of $X$ .

The definition, as one can see, is purely a set-theoretic one, with no requirement of a distance function / metric.

Clearly, every metric space $(X,d)$ naturally is a topological space $(X,\tau_d)$ with the topology $\tau_d$ induced from the metric $d$ . This metric-induced topology $\tau_d$ is nothing but the collection of all open subsets of the metric space $X$ , with respect to the metric $d$ .

Having defined the meaning of open sets, one can define ‘closed sets‘ of a topological space by defining them to be the complements of open sets:

A subset $F$ of $X$ is said to be closed if its complement $(X\setminus F)$ is open.

Now, if one is curious, it is natural to ask whether all topological spaces arise in the above manner, i.e. Given a general topological space, whether we can always find a metric whose metric-topology coincides with the given topology. This is called the metrizability problem for a topological space, as we are inquiring whether a given topological space is metrizable.
The answer, in general is ‘NO’. Not every topological space arises out of a metric. There are examples of non-metrizable topological spaces.

To begin with, let us recall the following:
In a metric space, if $x$ and $y$ are two distinct points, then there exist disjoint open sets $U$ and $V$ such that $x\in U$ and $y\in V$ ( This property is called the Hausdorff separation condition). $\dagger$

Hausdorff Separation Axiom

Now, let us take an example of a topological space defined in the following manner :
Take an infinite set $X$ and define the co-finite topology on $X$ by setting $\tau=\{A\subset X:A^c:=X\setminus A\text{\space is finite}\}$ . Now, take two distinct points $x$ and $y$ from $X$ .

Let, if possible, there be $U$ and $V$ , two open sets in $X$ , such that $x\in U$ , $y\in V$ and $U\cap V=\phi$ . By the definition of co-finite topology, $U=X\setminus\{x_1,\dots,x_n\}$ and $V=X\setminus\{y_1,\dots,y_m\}$ i.e. complements of finite sets in $X$ . Then $\phi=U\cap V=X\setminus\{x_1,\dots,x_n,y_1,\dots,y_m\}$ , implying that $X=\{x_1,\dots,x_n,y_1,\dots,y_m\}$ , which is a finite set, contradicting the fact that $X$ is infinite!

Thus, there are no such $U$ and $V$ which satisfy $x\in U,y\in V,U\cap V=\phi$ i.e. $X$ with the co-finite topology does not have the Hausdorff-separation property.

Thus, an infinite set equipped with the co-finite topology is not metrizable (Since metrizable spaces have Hausdorff-separation property by the observation $(\dagger)$ ).

So, we saw that not all topological spaces arise out of a metric / distance function.

In the upcoming posts, we shall try to define the analogous concepts from metric spaces, for a general topological space (e.g. limit points, closure, interior etc.).

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