Lecture 01

Lecture 01

Let be some set. denotes the set of all subsets of .

Let be called a topology on if:

1. .
2. is closed under arbitrary unions: if for all (where is an index set), then

3. is closed under finite intersections: if for some , then

The elements of are called the open sets of .

Examples and Non-examples

1. For any , let . This is the trivial topology on .

2. For any set , let . This is the discrete topology on .

3. Let .

a) . This is not a topology since

b) . This is a topology on .

c) . This is a topology on .

d) . This is not a topology since

Thus, the union of two topologies on need not be a topology.

e) . This is a topology on .

Finite Complement Topology

Let

Then (vacuously), and since is finite.

Suppose . We want to verify that

From the definition of , we need to be finite. By De Morgan’s law,

Since each , each is finite, so their intersection is finite. Hence

Similarly, one can prove that if for , then

Thus is a topology on .