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

Please state all definitions and theorems that you will need: D Let S be a subset of R . A point a: In of 17 such that N Q S. If for every ne then :r is called a boundary point of S . efim'tion 3.4.3 R is an interior point of S if there exists a neighborhogl N lghborhood N of a: , N F! S =, ill and N 0 (R\\S) The set of all interior points of S is denoted by mtS , and f the set of all boundary points of S is denoted by MS . Definition 3.4.6 Let S g R . If de Q S , then S is said to be closed. if de 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 S . (b) A set S is closed iff its complement R\\S is open. 2. The set of natural numbers,N, is: Open Closed Neither is} Explain: WTS N is closed: ' nENandnEN(n;5)foranynN _ i Since N(n; 6) (T N # 0 and N(n; 6) (R\\N) 0, i 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. '1 i i i 1 Since bd N = N , then by definition 3.4.6, since N contains all of its boundary, then N is closed. 2: WTS N is not open: Let N C; R , which consists only of natural num of N, a point, n E R is an interior point of N 1 . N (n; '2') Contains at least one number that 1 N (n; 5) that can be contained in the set does not have any interior points. It follows Since N does not have any interior points, .'V \".4