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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

- and
*X*are closed

*A*,*B*closed*A**B*is closed

- {
*A*_{i}|*i**I*} closed*A*_{i}is closed

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 *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 *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. and K2. follow from the definition.

To prove K3. note that *cl*(*A*) *cl*(*B*) is a closed set which contains *A* *B* and so *cl*(*A*) *cl*(*A* *B*).

Similarly, *cl*(*B*) *cl*(*A* *B*) and so *cl*(*A*) *cl*(*B*) *cl*(*A* *B*) and the result follows.

To prove K4. we have *cl*(*A*) *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**

- For
**R**with its usual topology,*cl*( (*a*,*b*) ) = [*a*,*b*] and*int*( [*a*,*b*] ) = (*a*,*b*).

- In the Sierpinski topology
*X*= {*a*,*b*} and ={ ,*X*, {*a*} }*cl*({*a*}) =*X*and*int*({*b*}) =

- If
*X*=**R**and = {,**R**} { (*x*, ) |*x***R**} then*cl*({0}) = (-, 0] and*int*( (0, 1) ) =

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