The existence of a limit point of a closed set

Question

Walter Rudin define a closed set as:

2.18 (d) A subset E of a metric space is closed if every limit point of E is a point of E.

I don't see in this definition nothing about the existence of a limit point of such set E. It's like the definition of a compact set, I don't see nothing about the existence of a open cover of such set, but I know that there is at least one open cover, its own metric space.

For me, the definition of a closed set doesn't guarantee the existence of a limit point, ¿am I right or not? and ¿Why?

Answer 1

The empty set is a closed set with no limit points. Other than that any discrete set also has no limit points. But, these are closed for the same reason the empty set is closed. Because they do contain all of their limit points, which is none.

Answer 2

No, the definition does not imply any limit points exist. If a set does not have any limit points at all, then all zero of its limit points are in the set and the set is closed. For example, the empty set is closed and the set of integers are closed.

Answer 3

You are right.

Consider this example:

Let $X = \{0 \}$, the set containing just one point: $0$.

Give $X$ the discrete topology $\mathcal{T} = \mathcal{P}(X) = \{ \emptyset, \{0\} \}$.

Then $X$ has no limit points (because there is a neighborhood of $0$ that contains no other points in $X$ but $0$ -- namely, $\{0\}$), but $X$ is certainly closed since the complement of $X$, which is $\emptyset$, is open.