# Commutative and Associative Addition in Closed Sets: A Conceptual Explanation

• tirwit
In summary, a vector space is a set of objects (vectors) that is closed under commutative and associative addition. This means that any result of an addition operation on two vectors must also be a vector in the set. This definition is different from the concept of closed sets in topology, where a set is considered closed if its complement is open. However, in a metric space, the two definitions are equivalent. The limit point characterization of closed sets is also equivalent to the closed under limit operation definition in a topological space that satisfies the first axiom of countability. Overall, closed sets in algebraic and topological contexts may have different definitions, but they both involve the idea of containment and closure.
tirwit
I'm reading Riley's "Mathematical Methods for Physics and Engineering" and I came across this expression about vector spaces:

"A set of objects (vectors) a, b, c, ... is said to form a linear vector space V if the set is closed under commutative and associative addition (...)"

What I don't understand is: what does commutative and associative addition have to do with a closed set?!

It doesn't have to do with closed sets.
In a topological space, a set is closed if the complement is open.

A space, V, is closed with respect to some operation, . , if for all x and y in V, we have that xy is in V. The definition of a vector space requires that all linear combinations of vectors is in the set, as well as a bunch of other axioms.

Take a look at the field axioms if you need some reference.

The essential point is that the term "closed" is in use in two completely different contexts.

In topology we talk about closed sets, which means sets containing all limit points - there may not be any arithmetic at all.

In arithmetic (generalized to such things as vector spaces) we mean that any result of an operation is contained within the space, while the topology is usually defined independently.

Ah ok ;)

Thanks both :)

mathman said:
The essential point is that the term "closed" is in use in two completely different contexts.

In topology we talk about closed sets, which means sets containing all limit points - there may not be any arithmetic at all.

In arithmetic (generalized to such things as vector spaces) we mean that any result of an operation is contained within the space, while the topology is usually defined independently.

that's not completely true since in a metric space a set is closed iff it's closed under "taking limits" i.e. closed with respect to convergent sequenes.

Riley is an excellent book by the way.

When saying that closure means that for a binary operation eg + on a set S

If A is in S and B is in S then A + B is in S

We mean for any finite number of such operations.

Taking an infinite number may result in another member of S or it may not. Many interesting cases that occur are the ones that do not.

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mathman said:
The essential point is that the term "closed" is in use in two completely different contexts.

In topology we talk about closed sets, which means sets containing all limit points - there may not be any arithmetic at all.

In arithmetic (generalized to such things as vector spaces) we mean that any result of an operation is contained within the space, while the topology is usually defined independently.

ice109 said:
that's not completely true since in a metric space a set is closed iff it's closed under "taking limits" i.e. closed with respect to convergent sequenes.
No, it's still "completely true". In a metric space, a closed set can be defined that way but in a metric space the two definitions are equivalent.

HallsofIvy said:
No, it's still "completely true". In a metric space, a closed set can be defined that way but in a metric space the two definitions are equivalent.

you've misunderstood what i meant.

mathman states that algebraic closure and set theoretic closure are two different things. i claim they're not that different. you claim what's completely true is my statement. hence contradiction.

A set C of the topological space X, is said to be closed if its complement, X-C, is open. (While a set is said to be open if it is an element of the topology T on X).

This is how Munkres defines it. A property/characterisitc of closed sets, as previously said, is that they contain all their limit points. Alternatively, the closure of a closed set is that set itself.

sutupidmath said:
A set C of the topological space X, is said to be closed if its complement, X-C, is open. (While a set is said to be open if it is an element of the topology T on X).

This is how Munkres defines it. A property/characterisitc of closed sets, as previously said, is that they contain all their limit points. Alternatively, the closure of a closed set is that set itself.

that's only true in metric spaces. in general topological spaces that doesn't necessarily make sense . hence the topological definition using complements.

ice109 said:
that's only true in metric spaces. in general topological spaces that doesn't necessarily make sense . hence the topological definition using complements.

I am not that sure what are you referring to here? But, if you are saying that closed sets do not contain their limit points in a general topological space, i will have to disagree with you...but again this might depend on how you define the limit point. In Munkres, he defines it this way: x is said to be a limit point of A, if every neighnorhood of x intersects A in a point other than x. With this definition in mind, then: a set is closed iff it contains all its limit points.

More importantly, if a topological space T satisfies the first axiom of countability (i.e. each point of T has a countable neighborhood base), then the limit point characterization of closed sets is equivalent to the closed under limit operation definition (closed iff every sequence in the space converges to a point in the space). Obviously getting rid of the metric doesn't mean that sequences are just useless, but the point ice109 was making regarding the closure of the limit operation still holds in a topological space if we impose additional conditions.

Anyways I'm pretty sure over half of these posts are completely useless to the OP, oh well.

ice109 said:
that's not completely true since in a metric space a set is closed iff it's closed under "taking limits" i.e. closed with respect to convergent sequenes.
"closed" in your statement is a particular example of the topological context. It doesn't mean that my assertions are not correct.

## What is the definition of a closed set?

A closed set is a subset of a topological space in which all of its limit points are contained within the set itself. In other words, a closed set is a set that contains all of its boundary points.

## What is the difference between a closed set and an open set?

A closed set includes all of its boundary points, while an open set does not. Additionally, a closed set is the complement of an open set, meaning that if a set is not closed, it is considered open.

## How can you determine if a set is closed?

To determine if a set is closed, you can check if all of its limit points are contained within the set. Alternatively, you can also check if the complement of the set is open. If the complement is open, then the set is closed.

## What are some examples of closed sets?

Some examples of closed sets include a finite set of numbers, a closed interval on the real number line, and a closed shape such as a circle or a square.

## Why are closed sets important in mathematics?

Closed sets are important in mathematics because they provide a way to define continuity and convergence. They also have applications in areas such as topology, functional analysis, and real analysis. Additionally, closed sets help to define other mathematical concepts such as closed maps and closed functions.

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