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

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*A set is closed if its complement with respect to X is open. I.e. {},{} and {,} are all open but not closed wrt $\tau_1$.

*I dont get what you mean by difference between $\tau_1$ and $\tau_2$ those are just families of subsets of X.

*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: 


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*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. 

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