ANALYSIS II

Metric Spaces: Open and Closed Sets

** Defn**
If
> 0, then an

** Defn**
A subset O of X is
called

** Proposition**
Each open -neighborhood
in a metric space is an open set.

** Theorem**
The following holds true for the open subsets of a metric space (X,d):

- Both X and the empty set are open.
- Arbitrary unions of open sets are open.
- Finite intersections of open sets are open. (Homework due Wednesday)

** Proposition**
Suppose Y is a
subset of X, and d

- (Y,d
_{Y}) is a metric space and open subsets of Y are just the intersections with Y of open subsets of X. - if Y is open in X, a set is open in Y if and only if it is open in X.
- in general, open subsets
relative to Y
**may fail**to be open relative to X.

** Examples**:
Arbitrary intersectons of open sets need not be open:

- If O
_{n }:= (-1/n, 1/n), then_{n}O_{n}= {0}. - If O
_{n}:= (-1/n, 1 +1/n), then_{n}O_{n}= [0,1).

** Defn**
Suppose X is a set and

- both X and the empty set belong
to
*T,* is closed under arbitrary unions,*T*is closed under finite intersections,*T*

then (X, ** T** )
is called a

** Defn**
A subset C of a metric space X is called

__Examples__:
Each of the following is an example of a closed set:

- Each closed -nhbd is a closed subset of X.
- The set {x
in
**R**| x d } is a closed subset of**C**. - Each singleton set {x} is a closed
subset of X.

- The
is a closed subset of*Cantor set***R**.

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

At the n-th stage, we have 2^{n}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 see that the Cantor set has many other interesting properties.

** Homework #4** (Due Monday
01/26)

- Suppose (X,
) is a topological space. Prove each of the following:*T* - Both
**X**and empty set are closed sets. - Arbitrary intersections of closed sets are closed.
- Finite unions of closed sets are closed.
- Show that {0 , 1 , 1/2 , 1/3 ,
... , 1/n , ...} is a closed set in
**R**and in**C**.