Defn.  A subset  O Í IR is called open if for each x ÎO there is an e-neighborhood N_e(x) contained in O.

1. Prove that each open interval (c,d) is an open set.

Theorem. The following holds true for open subsets of IR:

a.) Both IR and Æ are open.

b.) Arbitrary unions of open sets are open.

c.) Finite intersections of open sets are open.

a.) If O_n: = (-1/n, 1/n), then    Ç_{n = 1}^¥  O_n = {0}.

b.) If  O_n: = (-1/n, 1 +1/n), then   Ç_{n = 1}^¥  O_n = [0,1].

c.) If  O_n: = (-1/n, 1), then   Ç_{n = 1}^¥  O_n = [0,1).

a.) Both X and Æ belong to T,

b.) T is closed under arbitrary unions,

c.) T is closed under finite intersections,

Defn. A set C Í IR is called closed if its complement is open in IR.

Example. Each of the following is an example of a closed set:

a.) Each closed interval [c,d] is a closed subset of IR.

b.) The set (-¥,d ] : = {x Î IR| x £ d } is a closed subset of IR.

c.) Each singleton set { x₀} is a closed subset of IR.

d.) The Cantor set is a closed subset of IR.

 

To construct this set, start with the closed interval [0,1] and recursivley remove the open middle-third of each of the remaining closed intervals ...

At the n-th stage, we have 2ⁿ closed intervals each of length ([1/3])ⁿ:

Stage 0:                    [0,1]

Stage 1:        [0,[1/3]]                 [[2/3],1]

Stage 2:    [0,[1/9]]   [[2/9],[3/9]]       [[6/9],[7/9]]   [[8/9],1]

:

This finite union of closed intervals is closed. The Cantor set is the intersection of this (decreasing or nested) sequence of sets and so is also closed. Later, we will hopefully see that it has many other interesting properties.

Prove each of the following:

a.) Both IR and Æ are closed sets.

b.) Arbitrary intersections of closed sets are closed.

c.) Finite unions of closed sets are closed.

d.) {0,1,[1/2],[1/3], ..., [1/n], ... } is a closed set.