# First countable spaces and metric spaces.

• alemsalem
In summary, a first countable space is a topological space where every point has a countable neighborhood basis, meaning that there exists a countable collection of open sets containing the point. It differs from a metric space in that not all first countable spaces are metric spaces. Examples of first countable spaces include the real line, Euclidean spaces, and discrete spaces. They are important in mathematics for the development of concepts and theorems, and have real-world applications in fields such as physics and engineering, particularly in the study of chaotic systems and data analysis.
alemsalem

## Homework Statement

Show That every metric space is first countable. Hence show that every SUBSET of a metric space is the intersection of a countable family of open sets.

no equation

## The Attempt at a Solution

its easy to show that it is first countable, because for every point in the space there is the set of rational open balls which are included in every other open set.
but the second part of the question is confusing:
how can every subset be an intersection of a countable family? we only know that at every point there is a countable family but. there maybe an uncountable number of point..

Thanks for the help, I've been staring at the question for two hours..
by the way this problem is from P. Szekeres chapter 10 problem 10.9

Last edited:

To show that every subset of a metric space is the intersection of a countable family of open sets, we can use the fact that every metric space is first countable. This means that for every point in the space, there exists a countable family of open sets that contains that point.

Now, let S be a subset of a metric space X. For each point x in S, we can find a countable family of open sets that contains x. Since S is a subset of X, these open sets also contain all the points in S. Therefore, we can take the union of all these countable families of open sets to obtain a countable family of open sets that contains all the points in S.

Finally, we can take the intersection of this countable family of open sets to obtain the set S. This is because every point in S is contained in all the open sets in the countable family. Therefore, S is the intersection of a countable family of open sets, as required.

In conclusion, every subset of a metric space is the intersection of a countable family of open sets, using the fact that every metric space is first countable.

## 1. What is a first countable space?

A first countable space is a topological space in which every point has a countable neighborhood basis. This means that for each point in the space, there exists a countable collection of open sets that contain the point and any open set containing the point will also contain one of these sets.

## 2. How is a first countable space different from a metric space?

A metric space is a specific type of first countable space in which the topology is induced by a metric, or a distance function, on the space. However, not all first countable spaces are metric spaces, as the topology can also be induced by other types of functions.

## 3. What are some examples of first countable spaces?

Examples of first countable spaces include the real line, Euclidean spaces, and any finite or countably infinite discrete space. Other examples include the topological spaces that are induced by a metric, such as the Euclidean topology on a metric space.

## 4. Why are first countable spaces important in mathematics?

First countable spaces are important in mathematics because they allow for the development of important concepts and theorems, such as convergence and continuity, in a more general setting. They also provide a framework for studying more complicated spaces, such as manifolds, which often have a first countable topology.

## 5. How are first countable spaces used in real-world applications?

First countable spaces have many real-world applications, especially in fields such as physics and engineering. For example, they are used in the study of chaotic systems, where the behavior of a system can be described using a first countable space. They are also used in the design of efficient algorithms for data analysis and processing.

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