Convergence as for the cofinite topology on R

In summary: U$.The sequence $x_n \to x$ converges as for the cofinite topology iff for each open neighbourhood $U$ of $x$, $U = X \setminus \{s\}$ for a $s$ with $x \neq s$, it holds that almost all $x_n$ are in $U$. So if $x \neq s$, thenalmost all $x_n \neq s$. If infinitelymany $x_n=s$ then $x=s$.
  • #1
mathmari
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Hey! :giggle:

Does the sequence $x_n=\frac{1}{n}$ converges as for the cofinite topology on $\mathbb{R}$ ? If it converges,where does it converge? Could you explain to me what exactly is meant by "cofinite topology on $\mathbb{R}$" ? Do we have to define first this set and then check if we have convergence inside that set? :unsure:
 
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  • #2
mathmari said:
Does the sequence $x_n=\frac{1}{n}$ converges as for the cofinite topology on $\mathbb{R}$ ? If it converges,where does it converge?

Could you explain to me what exactly is meant by "cofinite topology on $\mathbb{R}$" ? Do we have to define first this set and then check if we have convergence inside that set?
Hey mathmari!

Wiki defines cofinite topology here.
It's the topology where every open set must either be the empty set, or it must have a finite complement. 🤔

Additionally we need the definition for convergence in a topology. We cannot use the usual $\varepsilon-\delta$ method, since distances are not defined in a topology.
Can we find that definition? 🤔
 
  • #3
Klaas van Aarsen said:
Wiki defines cofinite topology here.
It's the topology where every open set must either be the empty set, or it must have a finite complement. 🤔

Additionally we need the definition for convergence in a topology. We cannot use the usual $\varepsilon-\delta$ method, since distances are not defined in a topology.
Can we find that definition? 🤔

Do we use the following definition?

$\langle x_n:n\in\mathbb{N}\rangle$ converges to $x$ if and only if for each openneighborhood $U$ of $x$ there is an $m\in\mathbb{N}$ such that $x_n\in U$ whenever $n\ge m_U$.

:unsure:
 
  • #4
mathmari said:
Do we use the following definition?

$\langle x_n:n\in\mathbb{N}\rangle$ converges to $x$ if and only if for each openneighborhood $U$ of $x$ there is an $m\in\mathbb{N}$ such that $x_n\in U$ whenever $n\ge m_U$.
Yep. (Nod)
 
  • #5
Klaas van Aarsen said:
Yep. (Nod)

In general it holds that $\frac{1}{n}\rightarrow -$, so do we have to check if for each open neighborhood $U$ of $0$ there is an $m\in \mathbb{N}$ such that $\frac{1}{n}\in U$ whenever $n\geq m_U$ ?
Or do we apply the definition in practice? :unsure:
 
  • #6
mathmari said:
In general it holds that $\frac{1}{n}\rightarrow -$, so do we have to check if for each open neighborhood $U$ of $0$ there is an $m\in \mathbb{N}$ such that $\frac{1}{n}\in U$ whenever $n\geq m_U$ ?
Yep. (Nod)

What does an open neighborhood of $0$ look like? 🤔
 
  • #7
Klaas van Aarsen said:
What does an open neighborhood of $0$ look like? 🤔

Is it a ball with center the originand radius $\epsilon>0$ ? :unsure:
 
  • #8
mathmari said:
Is it a ball with center the originand radius $\epsilon>0$ ? :unsure:
Nope. (Shake)

We can only have a ball with a radius if we can measure distances. But in a topology those are not defined. Instead a neighborhood of a point is a subset that contains an open set - according to the topology. And moreover that open subset must contain the point. 🧐

Which open subsets are in the topology that contain 0? 🤔
 
  • #9
Klaas van Aarsen said:
Nope. (Shake)

We can only have a ball with a radius if we can measure distances. But in a topology those are not defined. Instead a neighborhood of a point is a subset that contains an open set - according to the topology. And moreover that open subset must contain the point. 🧐

Which open subsets are in the topology that contain 0? 🤔

So do we consider an interval around $0$ ? :unsure:
 
  • #10
mathmari said:
So do we consider an interval around $0$ ?

No, we have to apply the definition of a cofinite topology.
It says that the open subsets are the empty set plus all subsets that have a complement that is finite. 🤔
 
  • #11
mathmari said:
So do we consider an interval around $0$ ?

Btw, after we've established what a neighborhood of $0$ is in the cofinite topology, we will look at an interval around $0$ that is inside the neighborhood. 🤔
 
  • #12
Klaas van Aarsen said:
at does an open neighborhood of $0$ look like? 🤔

Is it an open set that contains an open subset containing 0? :unsure:
 
  • #13
mathmari said:
Is it an open set that contains an open subset containing 0?
More precisely, it's a subset $V$ of $\mathbb R$ that includes an open set $U$ containing $0$.
Note that $V$ is not necessarily open.
$$0 \in U \subseteq V \subseteq \mathbb R$$
🧐

Now what was an open set in the cofinite topology of $\mathbb R$ again? 🤔
 
  • #14
Klaas van Aarsen said:
Now what was an open set in the cofinite topology of $\mathbb R$ again? 🤔

It is a set that has finite complement or is empty, right? :unsure:
 
  • #15
Let $X$ be an arbitrary set. The non-empty open nsets are the complements offinite sets. We have to define also the empty set.

A sequence $x_n \to x$ converges as for the cofinite topology iff for each open neighbourhood $U$ of $x$, $U = X \setminus \{s\}$ for a $s$ with $x \neq s$, it holds that almost all $x_n$ are in $U$. So if $x \neq s$, thenalmost all $x_n \neq s$. If infinitelymany $x_n=s$ then $x=s$.

So in this case:

Let $U$ be an open neighbourhood of $0$. Since $\Bbb R\setminus U$ must be finite, $U$ containsallbut a fininte numberof terms of the sequence. Therefore $x_n\rightarrow 0$.

Is that correct? :unsure:
 
  • #16
mathmari said:
Let $U$ be an open neighbourhood of $0$. Since $\Bbb R\setminus U$ must be finite, $U$ contains all but a finite number of terms of the sequence. Therefore $x_n\rightarrow 0$.

Is that correct?
Looks right to me. (Nod)
So we conclude that $0$ is a limit of the sequence.
Is it 'the' limit? Can we tell for instance whether the sequence converges to $1$? (Wondering)
 
  • #17
Klaas van Aarsen said:
Looks right to me. (Nod)
So we conclude that $0$ is a limit of the sequence.
Is it 'the' limit? Can we tell for instance whether the sequence converges to $1$? (Wondering)

The limit is all the numbers of the form $\frac{1}{n}$ with $n\in \mathbb{N}$, right? :unsure:
 
  • #18
mathmari said:
The limit is all the numbers of the form $\frac{1}{n}$ with $n\in \mathbb{N}$, right?

Suppose we pick a number that is not of that form. Let's say we pick $\pi$.
Then an open neighborhood $U$ of $\pi$ is all of $\mathbb R$ except for a finite number of points, and it must include $\pi$ itself.
The neighborhood $U$ will contain all but a finite number of the sequence won't it? 🤔
 

1. What is convergence as for the cofinite topology on R?

Convergence as for the cofinite topology on R refers to the way that sequences of points in R behave when the topology on R is defined by the cofinite sets, meaning that the open sets are the complements of finite sets. In this topology, a sequence of points in R converges to a limit if and only if it contains all but finitely many points of the sequence.

2. How is convergence defined in the cofinite topology on R?

In the cofinite topology on R, a sequence of points (xn) converges to a limit L if for every open set U containing L, there is an index N such that for all n > N, xn ∈ U.

3. What are the properties of convergence in the cofinite topology on R?

Convergence in the cofinite topology on R has the following properties: it is unique, meaning that if a sequence converges to two different limits, then the limits must be equal; it is not Hausdorff, meaning that there can exist sequences that converge to multiple limits; and it is not first-countable, meaning that there is no countable basis for the topology.

4. How does convergence in the cofinite topology on R compare to other topologies on R?

Convergence in the cofinite topology on R is different from convergence in other topologies on R, such as the standard Euclidean topology. In the Euclidean topology, a sequence converges to a limit if and only if it eventually stays within any arbitrarily small neighborhood of that limit. In the cofinite topology, a sequence must contain all but finitely many points of the sequence to converge to a limit.

5. What are some applications of the cofinite topology on R?

The cofinite topology on R is often used in number theory and algebraic geometry, as it allows for a more natural definition of convergence for certain types of sequences. It is also useful in topology and analysis, as it provides a counterexample to some theorems that hold in other topologies, such as the Bolzano-Weierstrass theorem. Additionally, the cofinite topology has been used in computer science for data compression and coding theory.

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