# Is every Closed set a complete space?

• alemsalem
In summary, a closed set contains all its cluster points, while a complete set contains all limit points of Cauchy sequences. In a metric space, a closed interval can have points clustering without a cluster point, and completeness properties are not directly related to closed sets. A subset of a metric space is complete if and only if it is closed. The space \mathbb{R}\setminus \mathbb{Q} is an example of a closed set that is not complete. A space is complete metrizable if there exists a metric such that the space is complete, but not every metric makes the space complete. The space \mathbb{Q} is an example of a space that is not complete metrizable. A
alemsalem
a closed set contains all its cluster (accumulation) points: points for which any open neighborhood around them no matter how small contains points from the set.
a complete set contains all limit points of Cauchy sequences. which are very similar to cluster points.
My question is: in a metric space and a closed interval (usual topology) is it possible to have points clustering without a cluster point?

Closed sets and cluster points are concerns of topology and topology is independent of the chosen metric. Saying the space is metric does not factor into topological considerations. How familiar are you with the ideas of modern topology and manifolds?

I know they are topological considerations, but a metric space is a topological space and it has special properties, for example a metric space is "normal" which is not true for a general topological space. I am just wondering if in a metric space with a topology generated by open balls if closed sets and completeness properties are related . just like an open interval (EDIT: on the Real line) is not a complete metric space yet a closed interval would contain all its limit points..

I'm not so familiar, i just have been opening some books (M.Nakahara and P. Szekeres ) which are beyond my level, a lot of this I've seen yesterday for the first time..

Thanks!

The reverse is true independent of the chosen topology, while the doubt about the direct statement/question relies in the possibility that some Cauchy sequences wrt the metric topology are not convergent wrt the same topology. I'm sure a mathematician could come up with a Cauchy sequence which is not convergent wrt some metric topology, so that a closed set is not always a complete space.

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If a subset of a metric space is complete, then the subset is always closed. The converse is true in complete spaces: a closed subset of a complete space is always complete.

An example of a closed set that is not complete is found in the space $X=\mathbb{R}\setminus \mathbb{Q}$, with the usual metric. Then X is a closed set of itself but is not complete.
Curiously, there exists a metric on X such that X does become complete (and such that the topology doesn't change). So we say that X is complete metrizable (i.e. there exists a metric such that the metric space is complete, this does NOT mean that the space is complete for every metric).

One can say that a closed set of a complete metrizable space is complete metrizable. But there is not much more we can say.

A space that is not complete metrizable is $\mathbb{Q}$.

Wait a minute. By metric space, you mean Riemannian manifold, don't you? If not, then I have no idea.

No, by 'metric space' mathematicians mean <set endowed with a metric> in the sense of topology, not <manifold with a metric> in the sense of differential geometry.

## 1. What is a closed set?

A closed set is a subset of a metric space that contains all of its limit points. In other words, any sequence of points in the set that converges must also converge to a point within the set.

## 2. What is a complete space?

A complete space, also known as a Cauchy space, is a metric space in which every Cauchy sequence converges to a point within the space. In other words, there are no "missing" points in the space and all sequences converge.

## 3. Is every closed set a complete space?

No, not all closed sets are also complete spaces. A closed set may have missing points, or points that are not included in the set, making it incomplete.

## 4. Can a set be closed and incomplete?

Yes, a set can be closed but not complete. As mentioned before, a closed set may have missing points, making it incomplete. However, a set cannot be complete and not closed.

## 5. Why is it important to understand if a closed set is a complete space?

Understanding if a closed set is also a complete space is important in the study of metric spaces and topology. It helps us determine the properties and characteristics of different sets and spaces, and allows us to make predictions and proofs about them.

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