This question takes place in a general metric space $X$.
Let $x$ be an interior* point of $E \subset X$ iff there exists a deleted neighborhood of $x$ that is contained in $E$.
This is like the normal definition of "interior point", except it uses "deleted neighborhood" instead of "neighborhood", thus allowing a point not in $E$ to be an interior* point of $E$.
My question is: why is this not the standard definition of "interior point"? I see a couple reasons that it would make a more elegant system.
- "Limit point" and "interior* point" are both defined in terms of deleted neighborhoods ($x$ is a limit point of $E$ iff all deleted neighborhoods of $x$ include some point of $E$). This is more symmetrical.
- (Note: I do not yet have a general/categorical notion of duality) "Limit point" and "interior* point" are more adequately dual, for $x$ is a limit point of $E$ iff $x$ is not an interior* point of the complement of $E$, whereas this does not hold for "limit point" and "interior point".
- The dual notions of closure and interior are more symmetrically defined using "interior* point". The closure is defined as the union of $E$ and the set of limit points of $E$, and the interior is defined as the intersection of $E$ and the set of interior* points of $E$. The duality between closure and interior is harder to see with the standard definition of interior as the set of interior points of $E$. Also the proof that the complement of the closure of $E$ is the interior of the complement of $E$ reduces to a few applications of DeMorgan's law.
So why do people use "interior point" and not "interior* point"?