Question about axioms of topology

Question

In the three axioms of topology on $X$, denoted by $\tau_X$, which are,

1. $\emptyset \in \tau_X$, and $X \in \tau_X$
2. for elements(open sets) $u_1, u_2, ... \in$ $\tau_X$, $$\bigcup^{\infty}_{i = 1}u_i \in \tau_X$$
3. for elements(open sets) $u_1, u_2,... \in \tau_X$, $$\bigcap^{n}_{i=1}u_i \in \tau_X$$ What I want to ask is that why the third axiom of topology cannot be "the intersection of infinite number of elements are in topology class", is this due to some properties we want for topology, can somebody tell me why and if so, what will happen?

Answer 1

We want the intersection of open sets to open. However if you allow infinite intersections I can construct closed sets such as $\cap_{i=1}^\infty(-1-\frac1n,1+\frac1n) = [-1,1]$. This is why only finite intersections are allowed.

Answer 2

Let me first echo Brian Moehring in the comments and remind you that unions can be arbitrary, not merely countable.

Now, to answer your question, the short answer is that if we allow arbitrary unions and arbitrary intersections, then we often* get way too many sets for our topology to have any meaning. In fact, in the case of a metric space (or just the real line $\mathbb R$ if you like), once you decide that balls are open, allowing arbitrary intersections would mean that individual points were open (since each point is an intersection of all the balls centered around it), and then having arbitrary unions would tell you that literally every subset is open. At which point, openness becomes a completely meaningless concept.

So you can have arbitrary unions or arbitrary intersections, but not both. You might wonder, then, why do we do arbitrary unions and not intersections?

Returning to the metric space setting, we pointed out already that allowing arbitrary intersections would make individual points be open. Recall that $x_n\to x$ means eventually $x_n$ lies in every open neighborhood of $x$. So if $\{x\}$ is open, that means $x_n$ has to eventually be identically equal to $x$ just to converge. That's way too strong a requirement to do anything interesting.

*Update - as Charles Hudgins points out in comments, arbitrary unions and intersections are still studied in the form of so-called Alexandrov spaces. But as he also points out, we can’t assume points are closed in an Alexandrov space without the whole thing becoming discrete, so instead we get weird effects like the closure of a point containing other points, and sequences that converge to more than one point simultaneously.

This is, needless to say, a little confusing if you are trying to do analysis, for example.