# Topology: Understanding open sets

• nightingale123
In summary, the homework statement requires us to show that the open set ##\mathbb{N}^2## satisfies the following axioms for open sets:1) Every point which is not ##(0,0)## is open.2) Every union of such sets is open.3) There does not exist a sequence ##(x_{n})_{n\in\mathbb{N}}\subset\mathbb{N}^2 \text{ for which }\lim x_{n}\xrightarrow[n->\infty]{X}(0,0)##.I am having trouble visualizing the sets in ##\tau##, so I
nightingale123

## Homework Statement

We define ##X=\mathbb{N}^2\cup\{(0,0)\}## and ##\tau## ( the family of open sets) like this
##U\in\tau\iff(0,0)\notin U\lor \exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}##
##a)## Show that ##\tau## satisfies that axioms for open sets
##b)## Show that ##(0,0)## lies in the Closure of ##\mathbb{N}^2##
##c)## Describe closed sets in topology ##\tau##
##d)## show that there doesn't exists a sequence##(x_{n})_{n\in\mathbb{N}}\subset\mathbb{N}^2 \text{ for which }\lim x_{n}\xrightarrow[n->\infty]{X}(0,0)##.
Assume that X is not first-countable
##e)##

## The Attempt at a Solution

I'm having trouble visualizing the sets in ##\tau## I know from the first part that the every point which is not ##(0,0)## is open. Also I know that every union of such sets will also be open. Therefore ##\mathbb{N}^2## in itself is open. However I don't know how to visualize the other condition ##\exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}## I would really appreciate it if someone could explain to me how this sets look as I am unable to continue with the problem.
Thank you

What does ##\ni :## mean? And are you sure it is an OR in the defining property? That makes every set which doesn't contain ##(0,0)## an open set. Can you draw an open set which includes the origin, e.g. are ##\{(0,0),(1,1)\}## or ##\{(0,0),(1,\mathbb{N})\}## open?

nightingale123
https://i.gyazo.com/a809d7be047d7ed855211b4d51372367.png

This is the original picture of the problem and pretty I'm sure that I translated it correctly. The ##\ni## is the main part that is bothering me because I don't know exactly what it means

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Well, let's see:
So the open sets ##U ## are those which:
1) Do not contain ## (0,0) ## or :
2) The collection of pairs ## (n+k) \times \mathbb N - U ; k=0,1,2,...= \{ (n \times \mathbb N), (n+1) \times \mathbb N ,...,(n+k) \times \mathbb N ,... \} -U ## is finite . This means ##U## must contain EDIT all but finitely-many of the sets in each of the collections ## (n+k) \times \mathbb N; k=0,1,2,... ##
Does this help?

nightingale123
Best I can tell
This is the original picture of the problem and pretty I'm sure that I translated it correctly. The ##\ni## is the main part that is bothering me because I don't know exactly what it means[/QUOTE said:
...
It means something like " So that" , or " With the property that" . EDIT and so it seems like this may be:

https://en.wikipedia.org/wiki/Cofiniteness#Cofinite_topology

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nightingale123
Never seen it before. It probably means ##\forall ##. It's a bit strange and should be explained somewhere in the book.
Google translates "ali" by "but", so I assume it to mean "and" and not "or", which means that not all sets without the origin are open. Thus I read it as follows:
$$U \in \tau \Longleftrightarrow (0,0) \notin U\; \wedge \; \exists N \in \mathbb{N}\; \forall n > N \, : \,\vert \, \{n\}\times \mathbb{N} \backslash U \,\vert < \infty$$
I haven't checked whether it's a topology or not. I would feel more comfortable, if this guesswork above wouldn't had been necessary. Especially "ali" makes a major difference.

nightingale123
fresh_42 said:
Never seen it before. It probably means ##\forall ##. It's a bit strange and should be explained somewhere in the book.
Google translates "ali" by "but", so I assume it to mean "and" and not "or", which means that not all sets without the origin are open.
.
No, I think only those who contain ##(0,0)## and all-but-finitely-many elements in the set. Ze Risky topology you suggested in Random Thoughts. EDIT Then open sets are either those that contain all-but-finitely many elements of ##\mathbb N^2 ## or those that miss the origin. I think it is clear to see closedeness under union: 1) If neither set contains the origin, neither will the union; if each contains all-but-finitely-many, so will the union. 2) Clearly the whole space is there, as it contains all but finitely many ( none) . And clearly the empty set is also there. EDIT2: Still, don't b) and d) contradict each other?

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nightingale123
WWGD said:
No, I think only those who contain ##(0,0)## and all-but-finitely-many elements in the set. Ze Risky topology you suggested in Random Thoughts.
You are right and Google translate was wrong!
I checked two other translation pages and they both had "ali = OR"
Corrected condition:
$$U\in \tau \Longleftrightarrow (0,0)\notin U \vee \exists N\in \mathbb{N} \; \forall n>N\, : \, |\,\{n\}×\mathbb{N} \,\backslash \,U\,| < \infty$$

nightingale123
Thank you both
Now I tried to continue what WWGD said. A set open if it does not contain ##(0,0)## or there exists some number ##N## so that for every number greater than N ##(\{n\}\times\mathbb{N})\backslash U ## is finite.
So we need to check
##a)## whether any number of unions of such sets is again open
##b)## a finite number of intersection of such sets is again open
So checking for intersections:
Lets ignore the first part because if ##(0,0)\notin (A\wedge B)\implies (0,0)\notin A\cap B##
Now let's say that there exist two numbers ##N,M## for sets ##U,V##such that ##\forall n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}## and ##\forall n>M \implies(\{n\}\times\mathbb{N})\backslash V\text{ is finite}## if we define ##k=\max\{M,N\}## the set we again get is open because both ##U## and ##V## can only miss finite many points therefore their intersection can only miss finite many points.

I believe I understand now how this topology works and will probably be able to continue.

Thank you both very much for the help

WWGD

## 1. What is topology?

Topology is a branch of mathematics that studies the properties of geometric spaces that are preserved under continuous deformations, such as stretching or bending, but not tearing or gluing. It deals with the concept of open sets, which are sets that contain all the points inside them, but not on their boundaries.

## 2. How are open sets defined in topology?

In topology, open sets are defined as sets that contain all the points inside of them, but not on their boundaries. They are sets in which every point has a neighborhood that is also contained within the set. This is known as the "open set property".

## 3. What is the importance of open sets in topology?

Open sets are important in topology because they allow us to define the concept of continuity, which is a fundamental concept in topology. Continuity refers to the idea that small changes in the input of a function will only result in small changes in the output. Open sets also allow us to define other important concepts, such as compactness and connectedness.

## 4. How are open sets related to closed sets in topology?

In topology, open sets and closed sets are complementary concepts. A set is considered closed if its complement (the set of all points not contained in the set) is open. This means that a set can be both open and closed, or neither open nor closed.

## 5. What are some real-world applications of topology?

Topology has many real-world applications, such as in physics, biology, and computer science. In physics, topology is used to study the properties of space and matter. In biology, it is used to analyze the shape and structure of proteins and DNA. In computer science, topology is used to study networks and data structures, and to develop efficient algorithms for data analysis.

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