I'm new to the topology game and I think I'm understanding the general gist of these questions although I'm having some trouble with understanding some concepts
For set $S = \{(x,y) \in \mathbb R^2: 0 < x^2 + y^2 \le 1 \}$:
a) Find all interior points: would it be correct to say that; $0 < x^2 + y^2 < 1$ is all the points in the interior? since the original subset included the $1,$ and that means it's a boundary point, and not an interior point
b) For each interior point, find a value for $r$ for which the open ball lies within $S:$ in this case, $r$ denotes the radius right? so legitimate values are $0 < r < 1$
c) Find all boundary points: here all points on the curve $x^2 + y^2 = 1$ are boundary points
And finally...
d) is $S$ an open set, is it a closed set? Why or why not?
Here I'd say this is a closed set simply because of the inclusion of the boundary points, but obviously this is a really flimsy answer (along with the rest of my answers) and I was hoping someone could help legitimize them further
No, it is not correct to say 0 < $x^2 + y^2$ < 1 is all the interior points.
What is correct to say is that those points (x,y) for which $x^2 + y^2$ < 1 are the interior points.