Is intersection of two topologies on X a topology on X?

Answer 1

A topology is a set of elements or subsets that follows these properties:

• ∅ and X belongs to the set
• Any union of the subsets belongs to the set.
• Any intersection of the subsets belongs to the set.

Yes, any intersection of two topologies on X is always a topology on X. Consider this example:

Let X = {1,2,3}, T = {∅, {1},{2},{1,2},X} and S = {∅, {1}, {3}, {1,3},X}

Then, S ∩ T = {∅, X, {1}}

To show that S ∩ T is a topology, we need to prove these properties:

1. ∅ and X belongs to the set
2. Any union of the subsets belongs to the set.
3. Any intersection of the subsets belongs to the set.

Step 1: Prove the first property is followed

Since the empty set and X belongs to S ∩ T, the first property is followed. That is obvious. ;)

Step 2: Prove the second property is followed

Select any union of any pair of subsets. You should see that this property is also satisfied. How?

∅ U X = X ∈ S ∩ T

∅ U {1} = {1} ∈ S ∩ T

{1} U X = X ∈ S ∩ T

Step 3: Prove the third property is followed

Any intersection of the subsets belong to the set obviously. See below:

∅ ∩ X = ∅ ∈ S ∩ T

∅ ∩ {1} = ∅ ∈ S ∩ T

{1} ∩ X = {1} ∈ S ∩ T

So the intersection of two topologies on X is a topology.