Metric and Topological Spaces
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Properties of topological spaces

We can recover some of the things we did for metric spaces earlier.

Definition
A subset A of a topological space X is called closed if X - A is open in X.

Then closed sets satisfy the following properties

  1. empty and X are closed
  2. A, B closed implies A union B is closed
  3. {Ai | i belongs I} closed implies interoveriAi is closed

Proof
Take complements.

So the set of all closed sets is closed [!] under finite unions and arbitrary intersections.

As in the metric space case, we have

Definition
A point x is a limit point of a set A if every open set containing x meets A (in a point noteq x).

Theorem
A set is closed if and only if it contains all its limit points.

Proof
Imitate the metric space proof.


Definitions
The interior int(A) of a set A is the largest open set subset A,
The closure cl(A) of a set A is the smallest closed set containing A.

It is easy to see that int(A) is the union of all the open sets of X contained in A and cl(A) is the intersection of all the closed sets of X containing A.

Some properties

K1.  cl(empty) = empty
K2.  A subset cl(A) for any subset A
K3.  cl(A union B) = cl(A) union cl(B) for any subsets A and B
K4.  cl(cl(A)) = cl(A) for any subset A

Proof
K1. and K2. follow from the definition.

To prove K3. note that cl(A) union cl(B) is a closed set which contains A union B and so cl(A) subset cl(A union B).
Similarly, cl(B) subset cl(A union B) and so cl(A) union cl(B) subset cl(A union B) and the result follows.

To prove K4. we have cl(A) subset cl(cl(A)) from K2. Also cl(A) is a closed set which contains cl(A) and hence it contains cl(cl(A)).


Remark
These four properties are sometimes called the Kuratowski axioms after the Polish mathematician Kazimierz Kuratowski (1896 to 1980) who used them to define a structure equivalent to what we now call a topology.

Examples

  1. For R with its usual topology, cl( (a, b) ) = [a, b] and int( [a, b] ) = (a, b).
  2. In the Sierpinski topology X = {a, b} and curlyT ={ empty, X, {a} } cl({a}) = X and int({b}) = empty
  3. If X = R and curlyT = {empty, R} union { (x, infinity) | x belongs R} then cl({0}) = (-infinity, 0] and int( (0, 1) ) = empty

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JOC February 2004