Adherent Point, Accumulation Point, Boundary Point, Interior Point

Question

I'm drawing certain cases to better understand some of the jargon in introductory Topology. Can you guys quickly tell me if the four statements I made below are correct? I'm doubtful about the second statement the most.

Also is notion of accumulation points and adherent points generalizable to all topological spaces or like the definition states does it only hold in a Euclidean space? Thanks!

All definitions are relative to the space in which S is either open or closed below. These definitions are from Rudin's Principles of Math. Analysis

and these are from Apostol's Mathematical Analysis

Are the following statements true

1. In 2, if we take $S$ to be closed relative to $X$, then $x$ would be an interior point of S

2. In 2, this is the only case where $x$ is an adherent point but not an accumulation point

3. In 3, if we take the entirety of the neighborhood $N_{r}(x)$ and not just the shaded part then $x$ is not longer an interior point

4. The criteria for a boundary point is that any neighborhood of $x$ has nonempty intersection with either $S$ or $S^c$

Answer 1

1. Yes, $x$ is an interior point (of $S$). It is irrelevant if $S$ is open or closed, what you showed was that we could draw an open neighborhood around $x$ that was totally contained in $S$.
2. Based on the drawing, it seems that $x$ is an accumulation point of $S$, as for any open neighborhood you take around it (and remove $x$ from) contains a point of $S$. This is because it is on the boundary, so every open set around it contains some point in the set and outside it. Note that every accumulation point of aset has to be an adherent point(why?).
3. I think you may have this backwards. In its current state, $x$ is not an interior point (of the shaded area) but if you shade in the rest of the neighborhood, it will become one. This is because the way it is drawn, any open ball around $x$ will contain some point outside the shaded region and thus it cannot be an interior point.
4. This is correct. Note that there may seem to be connections between limit points and boundary points, but this is not always true.

Topology generalizes outside of Euclidean space, (and outside of metric spaces, where you define a distance function). It takes some work, but you can say sets are open and closed without using the concept of distance. You also have to adjust your definition of neighborhood slightly.