In the question, we are given a set $A \subseteq \mathbb{R}^n$ and need to decide whether each point in $A$ is an "inner" or "outer" point under a specific condition. The text states:

"Each point of $A$ is a [blank] point if and only if $A$ is open."

Answer

Each point of $A$ is an inner point if and only if $A$ is open. This is because in topology, a set $A \subseteq \mathbb{R}^n$ is open if, for every point in $A$, there exists an open neighborhood around it that is entirely contained within $A$. This makes each point in $A$ an inner point.

So, the correct term to fill in the blank is "inner" (innerer Punkt in German).

Would you like further details on the concept of inner points and open sets?

Here are 5 related questions to deepen understanding:

1. What is the definition of an open set in topology?
2. How does one determine if a point is an inner point of a set?
3. Can a set have both inner and boundary points? If so, when?
4. What is the difference between closed and open sets in $\mathbb{R}^n$?
5. How are inner points related to the concept of neighborhoods in topology?

Tip: In $\mathbb{R}^n$, every open set is a union of open balls, which helps in identifying inner points.