Could someone please check my work against the definitions I've provided

Please state all definitions and theorems that you will need: Definition 3.4.3 Let S be a subset of R . A point a in R is an interior point of S if there exists a neighborhood N of a such that NV C S. If for every neighborhood N of x , N n S # 0 and Nn (R\\S) # 0, then a is called a boundary point of S . The set of all interior points of S is denoted by intS , and the set of all boundary points of S is denoted by baS . Definition 3.4.6 Let S C R . If bdS C S , then S is said to be closed. If bdS C R\\S , then S is said to be open. Theorem 3.4.7 (a) A set S is open if S = intS . Equivalently, S is open iff every point in S is an interior point of (b) A set S is closed iff its complement R\\ S is open. -. ... 2. The set of natural numbers, N, is: Open Closed Neither Explain: WTS N is closed: n E N and n E N(n; E) for any n E N Since N(n; E) n N # 0 and N(n; E) n (R\\N) # 0, then by definition 3.4.3 for the boundary of N , N is in the boundary of N . Therefore, the boundary of N is N by definition 3.4.3. Since bd N = N , then by definition 3.4.6, since N contains all of its boundary, then N is closed. WTS N is not open: Let N C R , which consists only of natural numbers. By definition 3.4.3 for an interior point of N, a point, n E R is an interior point of N if some N(n; E) C N. Since every N n ; NIH contains at least one number that is not a natural number, then there is no N n; 1 that can be contained in the set N of strictly natural numbers. Therefore, N does not have any interior points. It follows that the interior of N is 0 . Since N does not have any interior points, then by definition 3.4.7(a), N is not open