Are Topological and Algebraic Closures Related?

[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?

 

[johnqwertyful]

Closed UNDER ADDITION or closed UNDER MULTIPLICATION.

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

 

[Bachelier]

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.

 

[micromass]

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.

Here is more information: http://en.wikipedia.org/wiki/Closure_operator

 

[Bachelier]

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.

Here is more information: http://en.wikipedia.org/wiki/Closure_operator

Cool..Thanks.

 

[Tac-Tics]

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).

 

[Number Nine]

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?

 

[Tac-Tics]

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.

 

[mathwonk]

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...