What is difference between closure of a set and closed set ?(Complex Analysis)

Question

I have read the following definition in Churchill complex variables and applications:

A set is closed if it contains all its boundary points. The closure of a set is a closed set consisting of all its points in S together with the boundary of S.

Closed set indeed will contain its points as well as its boundary points. Then what is the difference between a set being closed or a set being closure? What is the difference between the closure of a set and a closed set?

Answer 1

An arbitrary set is a closed set or is not a closed set. However, the closure of that set is always a closed set. (And if a set is a closed set, then the closure of that set is itself.)

For example, let $A$ be a set $\{x: |x| < 1\}$. $A$ is not a closed set. Because the set of boundary points of $A$ is $\{x: |x|=1\}$, and it is not contained in $A$. The closure of a set $A$ is $\{x : |x| \leq 1 \}$.

Answer 2

"A set being closure" is not a valid sentence.

If $X$ is the underlying set of your topology, then the closure is a mapping $\mathcal c\ell:\mathcal P(X)\to \mathcal P(X)$ such that $\mathcal c\ell(S)$ is the union of $S$ with its boundary for any $S\subset X$. Very often, we drop the formal function name and the closure function is indicated by $S\mapsto\overline S$.

By contrast, we say that a set $S\subset X$ is closed if it contains all of its boundary points.

These are obviously very tightly-related concepts. One might (and often does) say that a set $S$ is closed if it is its own closure, i.e. $S=\overline S$. Whether a set is $S$ closed or not, its closure is the intersection of all of the closed sets that contain $S$, i.e. $S=\bigcap\{T\subset X\mid S\subset T\text{ and $T$ is closed}\}$.

Answer 3

A set $A$ can be closed or not. If it is, it contains all its limit points. If it's not, we can add the "missing" limit points and that way form the closure of $A$, $\overline{A}$ and it turns out that this set is closed: we get no new limit points of $\overline{A}$ that weren't already limit points of $A$ (often stated as $A'' \subseteq A'$, which holds in $T_1$ spaces, in particular in metric spaces like $\Bbb C$).

So in this perspective a non-closed set is "incomplete", and we can enlarge it to it smallest closed superset (analogous to how we add reals to rationals to fill the "holes" in $\Bbb Q$, another completion process), its closure.

So linguistically: closed is a property that a set can have, the closure is the result of an operation on a set to make it closed by addition of points. (if a set is already closed, the closure operation does nothing, of course: $A = \overline{A}$ iff $A$ is a closed set).