Let's go through the definitions provided in the image with some examples to clarify each concept.

Definitions Breakdown with Examples

1. Neighborhood of $X_0$:

   • Definition: $E$ is a neighborhood of $X_0$ if there exists a radius $r > 0$ such that the open ball $B(X_0, r) \subseteq E$.
   • Example: Let $E = \{x \in \mathbb{R}^2 : |x| < 5\}$ and $X_0 = (1, 1)$. If you choose $r = 2$, then the ball $B((1,1), 2)$ is contained within $E$. Therefore, $E$ is a neighborhood of $(1, 1)$.

2. Interior Point of $E$:

   • Definition: $X_0$ is an interior point of $E$ if there exists an $r > 0$ such that $B(X_0, r) \subseteq E$.
   • Example: Consider $E = (0, 1)$ in $\mathbb{R}$. The point $X_0 = 0.5$ is an interior point because you can find a small interval around $0.5$, say $(0.4, 0.6)$, that is fully contained in $E$.

3. Exterior of $E$:

   • Definition: The exterior of $E$ is the set of all points that are interior points of the complement of $E$ and is denoted by $\text{Ext}(E)$.
   • Example: For $E = [1, 2]$ in $\mathbb{R}$, the complement $E^c$ is $(-\infty, 1) \cup (2, \infty)$. Any point in these intervals, like $X_0 = 0$ or $X_0 = 3$, is an interior point of $E^c$, so it belongs to $\text{Ext}(E)$.

4. Boundary Point of $E$:

   • Definition: $X_0$ is a boundary point of $E$ if every open set containing $X_0$ intersects both $E$ and its complement $E^c$.
   • Example: For $E = [1, 2]$ in $\mathbb{R}$, the points $X_0 = 1$ and $X_0 = 2$ are boundary points. Any open interval around these points will include values both inside and outside the set $E$.

5. Boundary of $E$ (denoted by $\partial E$):

   • Definition: The boundary of $E$ is the set of all boundary points of $E$.
   • Example: If $E = [1, 2] \cup (3, 4)$, then the boundary points are $\partial E = \{1, 2, 3, 4\}$ because these are the points where any open interval around them will contain elements both in $E$ and outside $E$.

Would you like more details on any of these concepts, or have any questions?

Further Questions to Consider:

1. How do we determine the interior of a set if it is defined by an inequality?
2. What is the difference between open and closed sets in the context of these definitions?
3. Can you provide examples of sets where the boundary is empty or trivial?
4. How does the concept of a neighborhood differ in other topological spaces?
5. What happens to the boundary of $E$ if the set $E$ itself is unbounded?

Tip:

When dealing with boundaries and neighborhoods, remember that the concept depends on the distance metric used. Different metrics can lead to different interpretations of these properties!