## Closed Sets Defined Relative to Properties of Open Sets in Topolgical Spaces

We form our topologies around the “basis” of open sets. However, we must also take a look at the closed sets that make the “holes” in our topology. So what is a closed set? A closed set is a set $A$ such that given an open set $X$ of a topology $\mathcal{T}_X$, we can say that $X-A$ is open. Let us now apply this to the real numbers $\mathbb{R}$. Our closed sets in this case are closed intervals $[a,b]$. This is valid because the complement $\mathbb{R}-[a,b]=(-\infty,a)\cup(b,\infty)$ which is an open set.

With this notion of closed sets in topologies, we may develop some conditions involve in any topological space $X$:

1. $\emptyset$ and $X$ are closed,
2. arbitrary intersections of closed sets are closed, and
3. finite unions of closed sets are closed.

We may prove this with DeMorgan’s law, however we will leave that to the reader as an exercise. However now we move on to what we call the interior and closure of a given subset $A$ of a topological space $X$. Essentially, the interior is the union of all open sets contained in $A$ and the closure is the intersection of all closed sets in $A$. We denote the interior of $A$ as $\text{Int} \ A$ and the closure of $A$ as $\text{Cl} \ A$. Automatically you should be thinking that $\text{Int} \ A$ is open and $\text{Cl} \ A$ is closed because the finite union of open sets are open and the arbitrary intersection of closed sets are closed. Therefore relative to $A$, we get

$\text{Int} \ A\subset A \ \text{Cl} \ A$.

Since we have been defining topologies through closed sets, we should define how bases would work. So given a topological space $X$ and a a subset $A$, we get the following properties pertaining to the basis of $X$:

1. $x\in\text{Cl} \ A$ iff every open set $U$ containing $x$ intersects $A$, and
2. $x\in\text{Cl} \ A$ iff every basis element $B$ containing $x$ intersects $A$.

And this brings back connections to our definition of a basis with respect to the open sets of the topological space $X$.