To show that the set of interior points of any set $E$, denoted as $\text{int}(E)$, is always open, we use the definition of an open set in a metric space or a topological space.

Proof:

Step 1: Definition of Interior Points

The interior of a set $E$, denoted by $\text{int}(E)$, consists of all points $x \in E$ such that there exists an open neighborhood $B_r(x)$ (a ball of radius $r$ centered at $x$) that is completely contained in $E$. Mathematically,

$\text{int}(E) = \{ x \in E \mid \exists r > 0 \text{ such that } B_r(x) \subseteq E \}.$

Step 2: Show that $\text{int}(E)$ is Open

To prove that $\text{int}(E)$ is open, we need to show that for every point $x \in \text{int}(E)$, there exists an $\varepsilon$-ball around $x$ that is entirely contained within $\text{int}(E)$.

• Let $x \in \text{int}(E)$. By definition, there exists some $r > 0$ such that the open ball $B_r(x) \subseteq E$.
• Consider any point $y \in B_r(x)$. We need to show that $y \in \text{int}(E)$, meaning there exists a small open ball around $y$ that is still in $E$.
• Define $\delta = r - d(x, y)$, which is positive because $y$ is inside $B_r(x)$.
• The ball $B_{\delta}(y)$ is entirely contained in $B_r(x)$, which is already inside $E$.
• Therefore, $y$ has an open neighborhood contained in $E$, implying $y \in \text{int}(E)$.

Since every point of $\text{int}(E)$ has an open neighborhood inside $\text{int}(E)$, it follows that $\text{int}(E)$ is an open set.

Conclusion

Thus, the interior of any set $E$ is always an open set.

Would you like a different explanation or more examples?

Related Questions:

1. What is the difference between the interior and closure of a set?
2. Can a set be equal to its interior? Give an example.
3. Is the interior of a closed set always empty?
4. How does the concept of interior relate to boundary points?
5. Prove that the interior of an intersection is a subset of the intersection of interiors.

Tip:

The interior of a set removes all boundary points and leaves only points that are "fully inside" the set. This is why it is always open!