The Interior Points of Sets in a Topological Space Examples 1

The Interior Points of Sets in a Topological Space Examples 1

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 called an interior point of $A$ if there exists an open set $U \in \tau$ such that:

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

We also proved some important results for a topological space $(X, \tau)$ with $A \subseteq X$:

• $A$ is open if and only if every $a \in A$ is an interior point of $A$, i.e., $A = \mathrm{int} (A)$.
• If $U \in \tau$ is such that $U \subseteq A$ then $U \subseteq \mathrm{int} (A)$.
• $\mathrm{int} (A)$ is the largest open subset of $A$.

We will now look at some examples regarding interior points of subsets of a topological space.

Example 1

Consider the set $X = \{ a, b, c \}$ and the nested topology $\tau = \{ \emptyset, \{ a \}, \{a, b \}, X \}$. Let $A = \{ a, c \} \subset X$. What are the interior points of $A$?

We note that all interior points of $A$ must be contained in $A$ by the definition of an interior point, so we need to only check whether $a \in A$ is an interior point and whether $c \in A$ is an interior point.

For $a \in A$, does there exists an open set $U \in \tau$ such that $a \in U \subseteq A$? Yes! The set $U = \{ a \} \in \tau$ and:

(2)
\begin{align} \quad a \in \{a \} = U \subseteq A = \{ a, c \} \end{align}

Therefore $a \in A$ is an interior point of $A$.

For $c \in A$, does there exist an open set $U \in \tau$ such that $a \in U \subseteq A$? No! The only set in $\tau$ containing $c$ is the wholeset $X = \{ a, b, c \}$ and $X \not \subseteq A$ since $b \in X$ and $b \not \in A$. Therefore $c$ is not an interior point of $A$.

Example 2

Consider an arbitrary set $X$ with the discrete topology $\tau = \mathcal P (X)$. Let $S \subseteq X$. What are the interior points of $S$?

Let $x \in S$. Since $S \subseteq X$, we have that $S \in \tau = \mathcal P(X)$. Let $U = S$. Then for each $x \in S$ we have that:

(3)
\begin{align} \quad x \in U = S \subseteq S \end{align}

Therefore every point $x \in S$ is an interior point of $S$.

Example 3

Consider an arbitrary set $X$ with the indiscrete topology $\tau = \{ \emptyset, X \}$. Let $S$ be a nontrivial subset of $X$. What are the interior points of $S$?

Let $S$ be a nontrivial subset of $X$. Then:

(4)
\begin{align} \quad \emptyset \subset S \subset X \end{align}

For all $x \in S$, we see from the nesting above that there exists no open set $U \in \tau$ such that $x \in U \subseteq S$. Therefore, every point $x \in S$ is not an interior point of $S$.