Difference between closed under countable union and arbitrary union

general-topology

From my research, I am confused in differentiating the difference between closed under arbitrary unions and countable unions.

  1. What exactly is the difference between them?

In an example, it states that:

Let $X$ be an uncountable set. Also let $\tau$ = {O $\subseteq X \mid$ O = X or O is at most countable}

For this example, without much explainantion, it states that $\tau$ is closed under finite intersections and under countable unions. But it isn't a topology on X as it isn't closed under arbitrary unions.

  1. How this example is not a topology?

My note on this is that since $\tau$ contains $\emptyset$ and $X$ so it has finite number of elements, hence it is countable. Taking their union will still be in $\tau$, so it is closed under countable unions. Why it is not closed under arbitrary unions? What's the difference?

Best Answer

First of all, I think you have some confusion about what $\tau$ is, based on your claim that $\tau$ is countable (finite, even!). $\tau$ is definitely uncountable: any uncountable set has uncountably many countable subsets - e.g. the set of singleton subsets is uncountable.

I think you may be conflating $\tau$ - which is a set of subsets of $X$ - with individual subsets of $X$.

Now as to the different types of union, it might help to consider a concrete example. Let $X=\mathbb{R}$, and consider the set $[0,1]$. Then:

  • $[0,1]$ is a union of sets in $\tau$, namely $$[0,1]=\bigcup_{x\in[0,1]}\{x\},$$ so $[0,1]$ had better be in $\tau$ if $\tau$ is to be a topology.

  • But $[0,1]$ isn't all of $X$ and it also isn't countable, so it's not in $\tau$. What's going on here is that while we can write it as a union of a bunch of sets in $\tau$ (as above), we can't write it as a union of only countably many sets in $\tau$.

The difference between countable unions and arbitrary unions is just how many sets we're allowed to "union together." In a countable union, we're taking the union of only countably many sets; in an arbitrary union, we're taking the union of as many sets as we want. For example, a countably union of countable sets is countable (briefly, "$\aleph_0\times\aleph_0=\aleph_0$" if you're familiar with $\aleph$-notation), but an arbitrary union of countable sets can be as big as you want: indeed, any set is an arbitrary union of one-element sets via $$S=\bigcup_{s\in S}\{s\}.$$

As a further exercise, thinking along the lines above we can show:

If $\tau$ is a topology on a set $X$ which contains every one-element subset of $X$, then in fact $\tau$ contains every subset of $X$ (we say $\tau$ is the discrete topology in this case).

Once you're comfortable with the proof of this fact, I think you'll understand the issues above perfectly.

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