sign in sign up
<<
Home , Isolated point, Limit point, R, X

Math 320 - October 21, 2020 11 Topology of the reals

Definition 11.1. Let ∈,>0. A neighborhood of x of size  is the (x,)={∈R:|x−y|<}. A deleted neighborhood of x of size  is the set N∗(x,)={y∈R:0<|x−y|<}. Clearly N∗(x,)=N(x,)\{x}. In our case of the set of real numbers R, N(x,)=(x−,x+), however the above definition of a neighborhood is more general and works for any metric spaces. Using the notion of a neighborhood, points in R can be classified as interior, or exterior to any particular subset ⊆R. Definition 11.2. Let S ⊆R. A x∈R is called interior to S, if there is a neighborhood of x that entirely lies in S, i.. x∈N ⊆S (∃>0,N(x,)⊆S). If for every neighborhood of x, N ∩S=6 ∅ and N ∩(R\S)=6 ∅, then x is called a boundary point for S, and x is called exterior to S, if there exists a neighborhood of x, such that N ∩S =∅. The set of interior points is denoted by int S, and the set of boundary points is denoted by bd S. Definition 11.3. A set S ⊆R is called an open set, if S =intS, i.e. all of its elements are interior points. To define what closed sets are, we first need the notion of a point. Definition 11.4. A point x∈R is called a or an accumulation point for S ⊆R, if every deleted neighborhood of x contains a point of S (∀>0,N∗(x,)∩S=6 ∅). The set of all accumulation points for S is denoted by S0. If a point of S is not in S0, then it is called an of S. One can show that the limit points of a set are precisely those that are the limits of convergent in the set. 0 Theorem 11.5. Let S ⊆R, a point s is a limit point of S, s∈S , if and only if there is a (sn)∈S, such that sn =6 s for all n∈N, and which converges to s, sn →s. Also, it’s not hard to see that the boundary of a set consists of limit points and the isolated points of the set. Using the notion of limit points, one can define closed sets as follows. Definition 11.6. A set S ⊆R is called a , if S0 ⊆S, i.e. it contains all of its limit points. Since the boundary points of a set are either the isolated points of the set, which are in the set by definition, or the limit points, then an alternative definition of a closed set would be the requirement that a closed set must contain its boundary points. One can also show the following theorem, which can act as an alternative definition of an open set, if the notion of a closed set is already defined, or an alternative definition of a closed set, if the notion of an open set is already defined. Theorem 11.7. A set S ⊆R is open, if an only if its complement, Sc =R\S, is closed. Combining open sets gives an open set, and intersecting closed sets results in a closed set. Theorem 11.8. (a) The union of any collection of open sets is open () The intersection of any finite collection of open sets is open. And as a direct corollary of the last theorem along with the preceding statement, gives the following. Theorem 11.9. (a) The intersection of any collection of closed sets is closed (b) The union of any finite collection of closed sets is closed. The union of S with the set of its accumulation points is called the of S, and is denoted by S =S∪S0. Using the closure of a set, one has an alternative characterization for closed sets (and hence, also for open sets by taking compliments). Theorem 11.10. Let S ⊆R. Then (a) S is closed iff S0 ⊆S (S contains all of its accumulation points) (b) S is closed () S is closed iff S =S () S =S∪bd S.