# Topology Example

I need to identify all possible topologies on the set X

$$ X = \{ a,b \} $$

To do this, I must consider all possible sets of open subsets that meet the definition of a topology.

**Topology Definition**. A topology on a set X is a collection T of subsets of X that includes the empty set ∅ and the set X itself, is closed under arbitrary unions of its members, and under finite intersections of its members.

For the set X={a,b}, the possible subsets are:

$$ P(X) = \{ ∅, \{ a \}, \{ b \}, X \} $$

Where X itself is the set {a,b}.

Therefore, in any topology T on the set X, the empty set ∅ and the set X itself, meaning X={a,b}, must always be included.

Now, I list all possible combinations of these subsets that satisfy the rules of a topology:

- The trivial (or minimal) topology, which includes only the empty set and the entire set: $$ T_1=\{ ∅, \{a,b \} \} $$
- The topology that includes, in addition to the sets of the trivial topology, the subset {a}: $$ T_2=\{ ∅, \{ a \} , \{a,b \} \} $$
- The topology that includes, in addition to the sets of the trivial topology, the subset {b}: $$ T_3=\{ ∅, \{ b \} , \{a,b \} \} $$
- The discrete topology (or maximal), which includes all possible subsets of the set X: $$ T_3=\{ ∅, \{ a \} , \{ b \} , \{a,b \} \} $$

These are **all the possible topologies on the set X**.

The trivial topology is the least restrictive, while the discrete topology is the most restrictive as it includes every possible subset of X as open.

In total, four topologies are possible on the set X.

### Example 2

Let's consider a set X consisting of three elements

$$ X = \{ a,b,c \} $$

I need to determine whether this collection of subsets is a topology on X

$$ T_3=\{ ∅, \{ a \} , \{ b \} , \{b,c \}, \{a,b,c \} \} $$

First, I check whether the collection includes both the empty set ∅ and the entire set X={1,2,3}

Both are present, so it passes the initial check.

Next, I examine if the collection is closed under the union of sets.

The collection T is not closed under union because the union {a}∪{b} results in the set {a,b} which is not included in T.

$$ \{ a \} \cup \{ b \} = \{a , b\} \ ∉ T $$

This alone confirms that the collection T does not form a topology on the set X.

It's completely unnecessary to check if the collection is closed under the intersection of sets.

And so on.