Example: In

, , and are clopen.]]>The open sets in are arbitrary unions of open intervals. The complement of is , a union of two open intervals; therefore its open (so is closed).

The open sets in

are more complicated to describe. Basically think of an open figure as a connected region of the complex plane that does not include the boundary. For example, the circle , which does not include points on the circumference. (Note that such a region need not be bounded; e.g. the half plane is an open figure.) Then the open sets in are arbitrary unions of open figures.All this may be very confusing for you, I know, but this is what

and are as topological spaces. It would be much simpler to treat them as metric spaces instead.]]>In

, an example of a closed set would be the interior and boundary of a circle (or square, or polygon, or any simply connected plane figure).]]>Thank you

Could you give an example of a closed set on the complex plane or the Real Number line?

]]>Let

and .Then

(a)

(its complement in is )(b)

(or any member of )(c)

and (these are always both open and closed in any topological space; they are called(d)

]]>(1) the union of any collection of sets that are elements of belongs to ;

(2) the intersection of any finite collection of sets that are elements of belong to ;

(3)the empty set and belong to .

Then, elements of are

A set

which is a subset of X isCould someone give me examples of

a) Closed Sets

b) Open Sets

c) sets which are both open and closed;

d) sets which are neither closed nor open.