Topological space - Infinite intersection of members

Question

The definition of Topological Space via open sets is :

A topological space is an ordered pair (X, τ), where X is a set and τ is a                   
collection of subsets of X, satisfying the following axioms:

-The empty set and X itself belong to τ.
-Any (finite or infinite) union of members of τ still belongs to τ.
-The intersection of any finite number of members of τ still belongs to τ.

Why the intersection of any infinite number of members of τ is neglected by the definition ?

To get things started: the general notions in topology should incorporate classical theory as a special case. In the real line, the infinite intersection of the open sets $(-1/n,1/n)$ is ${0}$ and, hence, not open. — Florian R, Aug 16 '18 at 14:34
Further, any closed interval $[a,b] \subset \mathbb{R}$ is equal to the intersection $\cap (a - \frac{1}{n}, b + \frac{1}{n})$ — Joe, Aug 16 '18 at 14:35
What do you think that an axiom should say about an intersection of an infinite number of members of $\tau$? — md2perpe, Aug 16 '18 at 15:18

Paul Frost · Answer 1 · 2019-10-16T13:35:43.583

Of course we are free to define what we want, and we could define a $\ast$-topological space as a pair $(X, \tau)$ satisfying the axioms of a topological space plus the stronger axiom that the intersection of any number of members of $\tau$ belongs to $\tau$.

However, this concept is not useful. For example, $\mathbb{R}$ with its usual topology (open sets are unions of open intervals) is not a $\ast$-topological space.

You probably know that topological spaces can alternatively introduced via closed sets, i.e. as pairs $(X, \alpha)$ where $\alpha$ is a collection of subsets of $X$ satisfying the following axioms:

The empty set and $X$ itself belong to $\alpha$.
Any intersection of members of $\alpha$ belongs to $\alpha$.
Any union of finitely many members of $\alpha$ belongs to $\alpha$.

Given $\alpha$, the set $\tau = \{ X \backslash A \mid A \in \alpha \}$ is a topology on $X$. Simililarly it works in the other direction.

In a $\ast$-topological space the union of any number of closed sets is a closed set. This is a very serious restriction. For example, if you want that all singletons $\{ x \}$, $x \in X$ are closed (which is the case in most topological spaces occurring in practical applicatons), then all subsets are closed which means that the space $X$ has the discrete topology (all sets are open).

Topological space - Infinite intersection of members

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