How does the topology of a space describe the closeness of the open subsets of a given set $X$?

Question

I've been trying to learn about topology recently and there is a thing that I couldn't understand. I know that given the topological space $(X,\tau)$, the $\tau$ contains open subsets of $X$. To my understanding, the $\tau$ exists to describe the closeness of the subsets of $X$ without using any kind of distance function (like in metric spaces). But how can we make these statements using the information given for a $\tau$.

E.g. let $(X,\tau_1)$ be a topological space with $X=\{a,b,c\}$ and $\tau_1=\{\emptyset,\{a,b,c\},\{a\},\{b\},\{a,b\}\}$. What kind of statements can we make about the closeness of the subsets $\{a\},\{b\}$ and $\{a,b\}$ (and about the element $c$)?
Now let $(X,\tau_2)$ be another topological space with the same $X$ but with the $\tau_2=\{\emptyset,\{a,b,c\},\{a\}\}$. What would be difference between $(X,\tau_1)$ and $(X,\tau_2)$?

Also another question: why do the elements of $\tau$ always have to be open sets? Why can't they be just closed? Or is it just the definition of the topolology which makes the subsets open?

Answer 1

About the last question: the choice to start with $\tau$ as the collection of open subsets is arbitrary. You could take the collection of closed subsets and you would still get the same information. This is because closed and open subsets are put in bijection by taking the complement. More precisely if you have $\tau$ a family of subsets satisfying the usual axioms of basis then $\{A^c : A \in \tau \}$ would give you a family of subsets satisfying a set of axioms complementary to the former (i.e. we ask now closure under arbitrary intersections and only finite unions).

Regarding the first and more interesting question: I think that the measure of how much two subsets are close is captured by their topological closure. In a metric space $(X,d)$ the usual definition of distance between two subsets $A,B$ is given by the formula $d(A,B)=\inf_{a \in A, b \in B}d(a,b)$. This in fact is the same as the distance between their closures $\overline{A}, \overline{B}$. Take as explicit example $\mathbb{R}$ with the usual distance: you have that a real number $r$ is intuitively close to a subset iff it has distance zero from a set. Let's take an interval $I=(a,b)$ with $b \leq r$: then the distance between $r$ and $I$ is given by $|b-r|$. Observe that the distance does not differentiate between the cases that $I$ is open or closed, in fact $d(I,r)=d(\overline{I},r)$.

The idea is that on a general topology $(X, \tau)$ you can distinguish two elements $x,y \in X$ if you can provide open subsets containing only one of the two: in the case of a metric space you have open balls giving you disjoint open neighborhoods separating the two, but in general spaces the situation is much more complicated and there are a lot of different separation axioms stating how much reasonably you can ask two points to be distinct.

Regarding the two examples of different topologies on the same set $X$: in the first case you can easily see that $\{c \}$ is closed while $\overline{\{a \}}=\{ a,c\}$ and similarly $\overline{\{b \}}=\{ b,c\}$. You have a situation where $c$ cannot be separated from the other elements since you do not have enough open subsets. For the second topology you have now $\overline{\{a \}}=X$: you could interpret this by saying that since you removed open subsets from your stating topology $\tau_1$ you have less information so you cannot distinguish $a$ from the rest of the set $X$ as well as before.

Answer 2

1. A set is closed if its complement with respect to X is open. I.e. {},{} and {,} are all open but not closed wrt $\tau_1$.
2. I dont get what you mean by difference between $\tau_1$ and $\tau_2$ those are just families of subsets of X.
3. As noted in 1. any set whose complement wrt X is open is called closed. You can define the same topology using its closed sets instead of its open sets.

I think its best to see "closed" and "open" as just arbitrary names for two reasons:

1. Topology aims at generalizing concepts thus the intuitive understanding we could try to carry over from $\mathbb{R^3}$ with a p-norm is not always helpful.

2. The English term "closed" is a poor and uninformative translation.