Understanding the Open Set: Definition & Examples

[hideelo]

What is the precise definition of the open set?

The definition I have been using until now has been that an open set is a set such that all of its points have some neighborhood that's contained in the set. The definition of neighborhood as far as I know is a collection of all the points within some given distance of a central point.

Now I know this cannot be correct since we do not need a set to be a metric space in order to define a topology. But if a topology is a collection of open sets, then how can you have that without having a defined distance function? Wouldn't you need a distance function to define a neighborhood, and without a neighborhood how can you define an open set?

 

[jgens]

An open set is just a member of the topology by definition. One can often provide examples of topologies without appealing to metrics. For example, let X be any set and let T = {ø,X}. Then T is a topology, so ø and X are the only open sets, and metrics were nowhere needed.

 

[hideelo]

Thanks for answering but I feel like this is getting circular. Is a topology a collection of open sets? Then how do you define open set?

Is an open set a member of a topology? Then how do you define topology?

 

[jgens]

Thanks for answering but I feel like this is getting circular.

The definition is not circular. One can define a topology without any reference to open sets.

 

[hideelo]

Then how do you define a topology? The definition I have is from "a first course in topology" by John McCleary where he says a topology is a collection of open sets satisfying certain conditions.

 

[economicsnerd]

By definition, a topological space is a pair $(X,\mathcal T)$, where:
-$X$ is a nonempty set.
-$\mathcal T$ is a collection of subsets of $X$ satisfying [insert axioms here].

The above definition makes sense, and it doesn't use the word "open" anywhere. So we're good.

Now, we define "open" to be a synonym for "$\in\mathcal T$". Said differently, given a subset $A\subseteq X$, the sentence, "$A$ is open," is taken to mean the exact same thing as "$A\in \mathcal T$."

 

[jgens]

Then how do you define a topology?

If X is a set, then a topology T is a collection of subsets of X, closed under arbitrary unions and finite intersections, which contains ø and X.

The definition I have is from "a first course in topology" by John McCleary where he says a topology is a collection of open sets satisfying certain conditions.

These sets turn out to be open a posteriori. There really is no circularity in the definition.

 

[pasmith]

Then how do you define a topology? The definition I have is from "a first course in topology" by John McCleary where he says a topology is a collection of open sets satisfying certain conditions.

McCleary is putting the cart before the horse.

A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ which satisfies certain conditions.

A subset $U \subset X$ is then defined to be open with respect to the particular topology $\mathcal{T}$ if and only if $U \in \mathcal{T}$.

If you have studied metric spaces you will have met "open balls" and a definition of an open set in terms of open balls. It is in fact the case that this definition is consistent with the above, because given a metric space one can always construct a topology on the underlying set in which the open sets are exactly those which satisfy the definition in terms of open balls.

 

[PeroK]

You can think of a topology as a generalisation of the concept open sets. You can define a topology on a set without a metric. The "open" sets are then, by definition, just the sets in the topology.

It also allows you to generalise the concept of continuity of a function without the need for a metric:

f is continuous if the pre-image of every set in the topology (open set) is in the topology (open set)

For a metric space (with the usual topology), this definition of continuity is equivalent to the normal definition using the metric and ε-δ.

 

[hideelo]

thanks so much to all of you, it is a real help. given the questionable nature of the book I'm using, can any of you recommend a good beginner book on topology?

thanks

 

[PeroK]

Hi I would look online. There are lots of maths notes floating about for free. Try looking for "introduction toplogy pdf" and see what comes up.

Also, even if you're working from a book, you can supplement what's in the book with online notes. Sometimes reading the same thing 2-3 different ways makes it click.

 

[jgens]

given the questionable nature of the book I'm using, can any of you recommend a good beginner book on topology?

The McCleary book looks fine. If you insist on alternatives, then the current standard text is Munkres. The GTM book by Kelley is supposed to be good as well.

 

[FactChecker]

What is the precise definition of the open set?

"Precise" is hard to define. For general topology work, the definition can be made in an abstract way by specifying the properties that you want for "open" sets (the union of open sets is open, the finite intersection of open sets is open, the null set and the entire set is open). You don't need a metric to define the concept of "open". You can see how much can be proven using only the minimal, abstract properties. You know that the conclusions hold for all situations where those minimal requirements are met. But if you are always working in a metric space, it is much more convenient to define open sets using the metric, and there is much more that you can say about those open sets.

 

[mathnerd15]

can you prove minimally that a set is open if its complement is closed or that a set is closed if it contains all of its limit points? what are some of the more beautiful and advanced uses of open sets?

 

[mathman]

can you prove minimally that a set is open if its complement is closed or that a set is closed if it contains all of its limit points? what are some of the more beautiful and advanced uses of open sets?

Your approach is becoming circular. Typically, in the abstract, a topology is defined by specifying a collection of subsets which obey certain rules (FactChecker post). Closed sets start out by being defined as complements of open sets.

 

[glb_lub]

thanks so much to all of you, it is a real help. given the questionable nature of the book I'm using, can any of you recommend a good beginner book on topology?

thanks

Try Topology by James Munkres.

 

[mathnerd15]

Your approach is becoming circular. Typically, in the abstract, a topology is defined by specifying a collection of subsets which obey certain rules (FactChecker post). Closed sets start out by being defined as complements of open sets.

so Rn is clopen because its complement the null set is open so it can be closed, while you can construct an open ball from any point which is interior to the set so that S=Int(S), all points are interior to the set since +- infinity aren't included in Rn?

is the null set really open since it contains no elements and Rn is both closed and open? maybe there's a problem with antinomy of the vocabulary?

 

[mathnerd15]

is there some contradiction where localized logical inferences don't hold true in other cases?

 

[Fredrik]

I guess I'm a bit late to the party, but I'd like to add how I like to approach these concepts. I like to define "closed" first.

A subset S of a metric space X is said to be closed if the limit of every convergent sequence in S is an element of S.​

I like this definition because the choice of the word "closed" is very natural if you're familiar with how it's used elsewhere in mathematics. For example a subset V of a vector space is said to be "closed under addition" if the sum of any two elements of V is in V. If you're familiar with that, and you're asked to guess what "closed under limits of sequences" means, wouldn't you guess exactly what's stated in the definition above? Then you should find the definition of "closed" very easy to remember, because a "closed set" is just a set that's closed under limits of sequences.

Now you can define the term "interior point" (I assume that you're already familiar with that definition) and prove the following theorem:

Let X be a metric space. Let S be a subset of X. The following statements are equivalent:

(a) X-S is closed.
(b) Every point of S is an interior point of S.
​

Then we state the definition of "open" as follows.

A subset S of a metric space X is said to be open if it satisfies the equivalent conditions of the theorem above.​

This approach explains why we use the term "open". Open sets are in a way the "opposites" of closed sets.

If we want to, we can include more conditions in the theorem, e.g.

(c) S is a union of open balls.​

The only reason I didn't is that I didn't want to use the word "open" until I had explained why it's natural to use it.

One of the most interesting things about open sets is that the definition of the "limit" of a sequence can be stated without explicit reference to the metric, by referring to open sets instead:

Let X be a metric space. Let S be a sequence in X. A point x in X is said to be a limit of S if every open set that contains x contains all but a finite number of terms of S.​

This suggests that if we may not need a metric at all to define limits of sequences. Maybe we can just choose a collection of subsets of some set X and just call those sets "open" subsets of X? That's the main idea behind the definition of "topological space".

However, we don't want to allow any collection of subsets to be labled as "open sets". We want the collection of sets that we call "open" to have some properties in common with the collection of open sets in an arbitrary metric space, because this will ensure that many theorems about metric spaces also hold for topological spaces. The properties that have been found to be the most useful are ones listed in the following theorem:

Let X be a metric space.

(a) ø and X are open.
(b) Every union of open sets is open.
(c) Every finite intersection of open sets is open.
​

(A finite intersection is an intersection of finitely many sets).

We are now ready to state the definition of "topological space":

Let X be a non-empty set. Let T be a set whose elements are subsets of X. The pair (X,T) is said to be a topological space if

(a) ø and X are elements of T.
(b) Every union of elements of T is an element of T.
(c) Every finite intersection of elements of T is an element of T.​

If (X,T) is a topological space, then T is said to be a topology on X.​

Now we can define the terms "open" and "closed" in the context of topological spaces:

Let (X,T) be a topological space. Let E be a subset of X. E is said to be "open" if E is an element of T. E is said to be "closed" if X-E is an element of T.​

 

[mathwonk]

this is a familiar phenomenon in mathematics. the most important example of open sets is the case where there is a metric. so if you understand that one you are in wonderful shape. now after that is understood, people began to try to abstract the concept and axiomatize it in order to make some more general analogous examples. The axioms that characterize a collection of "open sets" are the ones given above. but the intuition for these is also derived from the case of metric spaces.