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In topology we say that a set is $A$ is closed if its complement $A^\complement$ is open.

Now if we have a set $A$ and an operation $*$ we say that $A$ is closed under this operation if $x,y \in A$ then $x*y \in A$.

What is the relation between these two definition (if there is one )?

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1 Answer

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The idea of a "closure" is that we have some set that is incomplete in some way, and we want to supply the missing pieces.

For example, every natural number has a successor, a number that comes next. Is the set $\{0\}$ closed under successors? No, it's missing the successor of 0, which is 1. So put that in; now we have $\{0,1\}$. Is that closed? No, now it's missing the successor of 1. But we can ask for the smallest set that is closed under the successor operation: it is the set of natural numbers. ( The empty set is also closed, but the natural numbers is the smallest closed set containing 0.)

Similarly, consider the operation of taking limits. In the rational numbers, not every convergent (Cauchy) sequence has a limit. If we consider the closure under this operation, and add in the limit points, we get its closure, the real numbers, which is the smallest closed set containing the rationals. There are other closed sets containing the rationals, but the reals are the smallest one.

In topology we have a closely related notion of closure. A set $S$ may have some “limit points”, and the closure $\text{cl}(S)$ is the smallest set that contains $S$ and all its limit points. It so happens that $\text{cl}(\text{cl}(S)) = \text{cl}(S)$, so that we don't have to take the process any further as we did when constructing the natural numbers.

Now let's consider a geometric object like a cube. It has some symmetries: for example if we rotate it a quarter turn in some direction, it looks the same. Let's call this $r$. Composition of two symmetries (do one, then the other) produces a new symmetry. Is the set $\{r\}$ closed under composition? No, it's missing $r^2$, which is the half-turn rotation. Is $\{r, r^2\}$ closed? No. But $\{r^0, r^1, r^2, r^3\}$ is closed. It's the smallest set of symmetries that contains $r$ and is closed under composition. A set of symmetries closed under composition is called a "group of symmetries" or just a "group", and the closure of a set of symmetries $S$ is "the group generated by $S$". (I am simplifying a little. For groups the operation is composition or taking inverses, but for finite groups you get the inverses for free.)

The span of a set of vectors is the smallest closed set containing those vectors. The closure is over the operation of taking linear combinations. What is a vector space? Nothing more nor less than a closed set of vectors.

Closures are everywhere in mathematics.

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