Are Topological and Algebraic Closures Related?

• Bachelier
In summary, the term "closed" in mathematics refers to the property of a set being closed under a specific operation. In topology, a set is closed if it contains all of its limit points. In algebra, closure under an operation means that the result of the operation is still within the same set. While these concepts may seem similar, they are not necessarily related. However, there is some vague relation between closed sets in topology and algebra, as they both have similar properties such as being closed under certain operations. The term "closure" is often used to describe this property.
Bachelier
In topology, when we say a set is closed, it means it contains all of its limit points

In Algebra closure of S under * is defined as if a, b are in S then a*b is in S.

Are these notations similar in any way?

Last edited:
Closed UNDER ADDITION or closed UNDER MULTIPLICATION.

Vs just closed. Don't think they're necessarily related.

johnqwertyful said:
Closed UNDER ADDITION or closed UNDER MULTIPLICATION.

Vs just closed. Don't think they're necessarily related.

Or closed under the group operator *.

So they are not related. The question just crossed my mind.

There is some vague relation between the two.

For example, given a group G, you can look at all the sets closed under *. Call $\mathbb{C}$ the sets closed under the multiplication. Then we have some eerily familiar properties:

• $\emptyset, G\in \mathcal{C}$
• If $C_i\in \mathcal{C}$ for all $i\in I$, then $\bigcap_{i\in I} C_i\in \mathcal{C}$

The only difference here is that the union of two sets in $\mathcal{C}$ need not be in $\mathcal{C}$.

So you see that the closed sets in topology and the closed sets in algebra have some quite similar properties.

micromass said:
There is some vague relation between the two.

For example, given a group G, you can look at all the sets closed under *. Call $\mathbb{C}$ the sets closed under the multiplication. Then we have some eerily familiar properties:

• $\emptyset, G\in \mathcal{C}$
• If $C_i\in \mathcal{C}$ for all $i\in I$, then $\bigcap_{i\in I} C_i\in \mathcal{C}$

The only difference here is that the union of two sets in $\mathcal{C}$ need not be in $\mathcal{C}$.

So you see that the closed sets in topology and the closed sets in algebra have some quite similar properties.

Cool..Thanks.

In algebra "closed under the operator *" and the like are actually redundant. If you consider an operator as a function, it's automatically closed on its domain.

(Just be wary of things like division. On the real numbers / is actually defined as a function R x (R-{0}) -> R).

Tac-Tics said:
In algebra "closed under the operator *" and the like are actually redundant. If you consider an operator as a function, it's automatically closed on its domain.

It's not redundant at all. How would you define the notion of a subgroup (or any other sub-object) without mentioning closure?

Number Nine said:
It's not redundant at all. How would you define the notion of a subgroup (or any other sub-object) without mentioning closure?

I could see how the word closure is still useful there.

Just to be clear, though, in the definition of a group on its own, it is strictly redundant:

Let G be a set and * : G x G -> G be an associative function such that there is an element e ∈ G such that e * x = x and x * e = x, and for each x, there is a y such that x * y = e and y * x = e.

The fact that the domain is G x G and the codomain of * is G implies that * is closed.

For a subgroup (U, **), you have to show that ** a "subfunction" of * with type U x U -> U. I'll concede using the term closure is a concise way of doing this.

the word "closed" generally means that the result of performing a certain operation lands you back in the same set you started in. The operation referred to can vary. In algebra the operation is addition or multiplication or whatever, and in topology it means taking limits. so an additive submonoid is closed under taking sums, an additive subgroup is closed under sums and differences, a closed set in a topological space is closed under taking limits...

What is closure in topology and algebra?

Closure in topology and algebra refers to a mathematical concept that describes the smallest set that contains all the elements of a given set. It is often denoted as "cl(A)" and can be thought of as the "completeness" of a set.

What is the difference between closure and boundary?

Closure and boundary are two related concepts in topology and algebra. The closure of a set includes all the points in the set, as well as any limit points that may be outside of the set. The boundary of a set refers to the points that are on the edge or "boundary" of the set, and are not included in the set itself.

How is closure used in real-world applications?

Closure has many practical applications in fields such as engineering, physics, and computer science. In engineering, closure is used to analyze the stability of structures and systems. In physics, closure is used to describe the completeness of a physical system. And in computer science, closure is used to define functions and data structures.

What are some properties of closure?

Some key properties of closure include: it is always a closed set, it is the smallest closed set containing the original set, it is idempotent (i.e. applying closure twice yields the same result), and it satisfies the Kuratowski closure axioms.

Can closure be applied to infinite sets?

Yes, closure can be applied to infinite sets. In fact, it is often used in topology and algebra to describe infinite sets and their completeness. However, it is important to note that in some cases, the closure of an infinite set may be finite or empty.

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