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.