A Comparison of the Interior and Closure of a Set in a Topo. Space

A Comparison of the Interior and Closure of a Set in a Topological Space

Recall from The Interior Points of Sets in a Topological Space page that if $(X, \tau)$ is a topological space and $A \subseteq X$ then a point $a \in A$ is said to be an interior point of $A$ if there exists a $U \in \tau$ with $a \in U$ such that:

(1)
\begin{align} \quad a \in U \subseteq A \end{align}

We called the set of all interior points the interior of $A$ and denoted it by $\mathrm{int} (A)$. We also saw that the interior of $A$ is the largest open set contained in $A$, i.e., $\mathrm{int} (A) \subseteq A$.

On The Closure of a Set in a Topological Space page we saw that if $(X, \tau)$ is a topological pace and $A \subseteq X$ then the closure of $A$ denoted $\bar{A}$ is the smallest closed set containing $A$, i.e., $A \subseteq \bar{A}$.

You may have noticed that the interior of $A$ and the closure of $A$ seem dual in terms of their definitions and many results regarding them. Consequentially, we will compare both of these sets below.

A Comparison of the Interior and Closure of a Set

The Interior of $A$, $\mathrm{int} (A)$ The Closure of $A$, $\bar{A}$
The interior of $A$ is the LARGEST OPEN set CONTAINED in $A$ The closure of $A$ is the SMALLEST CLOSED set CONTAINING $A$
$\mathrm{int} (A) \subseteq A$ $A \subseteq \bar{A}$
$A$ is OPEN if and only if $A = \mathrm{int} (A)$ $A$ is CLOSED if and only if $A = \bar{A}$
If $A \subseteq B$ then $\mathrm{int} (A) \subseteq \mathrm{int} (B)$ If $A \subseteq B$ then $\bar{A} \subseteq \bar{B}$
$\mathrm{int} (A) \cup \mathrm{int} (B) \subseteq \mathrm{int} (A \cup B)$ $\bar{A} \cup \bar{B} = \overline{A \cup B}$
$\mathrm{int} (A) \cap \mathrm{int} (B) = \mathrm{int} (A \cap B)$ $\bar{A} \cap \bar{B} \supseteq \overline{A \cap B}$

A good way to remember the inclusion/exclusion in the last two rows is to look at the words "Interior" and Closure.

The intersection of interiors equals the interior of an intersection, and the intersection symbol $\cap$ looks like an "n".

The union of closures equals the closure of a union, and the union system $\cup$ looks like a "u".

If $(X, \tau)$ is a topological space and $A \subseteq X$, then it is important to note that in general, $\mathrm{int} (\overline{A})$ and $\overline{\mathrm{int}(A)}$ are different.

Example 1

Consider the $\mathbb{R}$ with the usual Euclidean topology and let $A = \mathbb{Q}$ (the set of rational numbers). Then $\mathrm{int} (\mathbb{Q}) = \emptyset$, while $\overline{\mathbb{Q}} = \mathbb{R}$. Therefore we see that:

(2)
\begin{align} \quad \mathrm{int} (\overline{\mathbb{Q}}) = \mathbb{R} \end{align}

And meanwhile:

(3)
\begin{align} \quad \overline{\mathrm{int} (\mathbb{Q})} = \emptyset \end{align}

So order is very important.