Help me understand how we define closed sets in real analysis

Question

I'm currently reading the second version of Understanding Analysis by Stephen Abbott and on page $90$ theorem $3.2.8.$ states that "A set $F \subseteq \mathbb{R}$ is closed if and only if every Cauchy sequence contained in $F$ has a limit that is also an element of $F$". According to definition $3.2.7.$ on the same page "A set $F \subseteq \mathbb{R}$ is closed if it contains its limit points".

The proof was left as an exercise. I think I pretty much proved it but I need help understanding some of the logic involved here.

If $F$ only contains a single point then there are no sequences in $F$ that do not contain their limit so there are no limit points. Is $F$ still closed? I assume that some sort of logical mambo jambo can be applied here to show that this is the case. To say that $F$ contains all its $0$ limit points does not sit very well with me since we're treating "nothing" as a "something". We're essentially saying that $F$ contains this "nothing". It makes no sense to me. Personally I decided to just embellish the definition with "If $F$ only contains one point then it is closed, otherwise blablabla...". Is this equivalent to the previous definition?

Answer 1

You are correct when you say that a singleton set does not contain any limit points.

However, think of the proposition in the following fashion : $F$ is closed, if the set of limit points of $F$ is a subset of $F$.

Then, the set of limit points of a singleton set, is the empty set, and is therefore contained in $F$. This notion clarifies the doubt of how $F$ can contain nothing, to a better notion of set containment.

Answer 2

Note that a constant sequence is a Cauchy sequence. The point is that a closed set will have two type of Cauchy sequences:

1. Those that converge to a limit point of the set,
2. and those which are converge to an isolated point of the set (these will be eventually constant sequences).

So $\{0\}$ is a closed set, because the only Cauchy sequence is converging to $0$, and that's fine.

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What this is telling us is that there are two type of points in a closed set: limit points, and isolated points. And Cauchy sequences can converge to either one, only that in the latter case the sequence has to be eventually finite.

As for your embellishment, not that there is no actual difference between a singleton and a finite set. Or $\Bbb Z$, or any closed and discrete set (i.e. a closed set of isolated points). It is a good thing to try and play with definition in order to understand them, but it's important to remember a few test cases and not just the basic one (the classic ones would be: empty, singleton, finite, discrete, dense, open, closed, and arbitrary), so before you come up with revised definitions, it is often a good idea to try a few cases and not just singletons.