Basic topology definition: open sets as arbitrary unions but only finite intersections? Why is it that in topology one can obtain an open set by taking the union of arbitrarily many open sets, but if one wishes to obtain an open set from intersections, one is restricted to taking only finitely many intersections?
I saw the answers to this question but I'm still confused:
On definition of a topology
Thank you very much.
 A: Perhaps an example will help. Consider the standard $\mathbb{R}$. You may be aware that intervals of the form $(a,b)$ are open. But intervals of the form $[a,b]$ are not, these are closed. In particular singletons $\{a\}$ are not open.
So consider now open intervals $U_n=(-1/n, 1/n)$ for any $n\in\mathbb{N}$. You can then check that $\bigcap_{n=1}^\infty U_n=\{0\}$ which is not open.
Now of course you can invent a new definition such that an arbitrary intersection of open subsets is again open. And in fact we do consider such situations. This is also known as the Alexandrov topology. So $\mathbb{R}$ is not Alexandrov, but for example every topological space that is finite as a set is Alexandrov. But this situation is rather rare in practice and less useful then the classical definition of topology. For example $T_1$ Alexandrov space is automatically discrete. And 99% of spaces we encounter "in the wild" are non-discrete $T_1$.
We can invent definitions however we want. But the rule of thumb is: a definition is as it is, because it is useful.
