# Closed and Open Subsets of a Metric Space

• gajohnson
In summary, the conversation discusses a proof that a given set with a specific metric is indeed a metric, and explores the properties of open and closed subsets of this set with the given metric. It is concluded that every subset of the set is both open and closed.
gajohnson

## Homework Statement

Let $X$ be an infinite set. For $p\in X$ and $q\in X$,

$d(p,q)=1$ for $p\neq q$ and $d(p,q)=0$ for $p=q$

Prove that this is a metric. Find all open subsets of $X$ with this metric. Find all closed subsets of $X$ with this metric.

NA

## The Attempt at a Solution

I showed easily that this is indeed a metric.

On the second part of the question, it seems to be the case that all subsets $\left\{x\right\}$ for all $x\in X$ are open because choosing a radius less than 1 gives a neighborhood around $x$ which only contains $x$ itself.

But then any subset of $X$ should be open, shouldn't it? Because each point of that subset can be shown to be an interior point using the logic above.

Similarly, there should be no closed subsets. Each point in a subset of $X$ obviously has a neighborhood which contains only that point.

Any ideas? Thanks!

gajohnson said:

## Homework Statement

Let $X$ be an infinite set. For $p\in X$ and $q\in X$,

$d(p,q)=1$ for $p\neq q$ and $d(p,q)=0$ for $p=q$

Prove that this is a metric. Find all open subsets of $X$ with this metric. Find all closed subsets of $X$ with this metric.

NA

## The Attempt at a Solution

I showed easily that this is indeed a metric.

On the second part of the question, it seems to be the case that all subsets $\left\{x\right\}$ for all $x\in X$ are open because choosing a radius less than 1 gives a neighborhood around $x$ which only contains $x$ itself.

But then any subset of $X$ should be open, shouldn't it? Because each point of that subset can be shown to be an interior point using the logic above.

Correct. All sets are open.

Similarly, there should be no closed subsets.

You might want to rethink this.

micromass said:
Correct. All sets are open.

You might want to rethink this.

OK, so every subset of $X$ contains no limit points, so every subset of $X$ must be closed.

...so every subset of $X$ is both open and closed?

That is correct!

micromass said:
That is correct!

Excellent. Many thanks to you!

## 1. What is a metric space?

A metric space is a mathematical structure that consists of a set of points and a distance function. The distance function, also known as a metric, measures the distance between any two points in the space.

## 2. What is a closed subset?

A closed subset of a metric space is a subset that contains all of its limit points. In other words, any sequence of points in the subset that converges to a point outside of the subset is not considered a limit point.

## 3. What is an open subset?

An open subset of a metric space is a subset that does not contain any of its boundary points. In other words, all points in the subset are considered interior points.

## 4. How do closed and open subsets relate to each other?

Closed and open subsets are complements of each other. This means that if a subset is closed, then its complement is open, and vice versa. Additionally, any set can be written as a union of a closed subset and an open subset.

## 5. Why are closed and open subsets important in metric spaces?

Closed and open subsets are important because they help us understand the structure of a metric space. They also play a crucial role in defining continuity and other topological properties in metric spaces.

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