Topologies on a Finite 3-Element Set

Topologies on a Finite 3-Element Set

Recall from the Topological Spaces page that a set $X$ an a collection $\tau$ of subsets of $X$ together denoted $(X, \tau)$ is called a topological space if:

• $\emptyset \in \tau$ and $X \in \tau$, i.e., the empty set and the whole set are contained in $\tau$.
• If $U_i \in \tau$ for all $i \in I$ where $I$ is some index set then $\displaystyle{\bigcup_{i \in I} U_i \in \tau}$, i.e., for any arbitrary collection of subsets from $\tau$, their union is contained in $\tau$.
• If $U_1, U_2, ..., U_n \in \tau$ then $\displaystyle{\bigcap_{i=1}^{n} U_i \in \tau}$, i.e., for any finite collection of subsets from $\tau$, their intersection is contained in $\tau$.

We will now look more into some of the topologies that can be applied on a finite $3$-element set, $X = \{a, b, c \}$. There are many different topologies that can be constructed on $X$.

For example, the indiscrete topology containing only the empty set and $X$ itself, $\tau_1 = \{ \emptyset, X \}$ is illustrated below:

For another example, $\tau_2 = \{ \emptyset, \{ a \}, X \}$ is a topology and is illustrated to be:

We can also construct a topology $\tau_3 = \{ \emptyset, \{b, c \}, X \}$ which looks like:

Or even a topology $\tau_4 = \{ \emptyset, \{ a \}, \{ b, c \}, X \}$ looking like:

There are a ton of other topologies that can be constructed with this small finite set $X$ - so you may be wondering, is every collection of subsets from $X$ a topology? The answer is no. For example, consider the collection $\rho = \{ \emptyset, \{ a \}, \{ b \}, X \}$ is not a topology on $X$ because the union $\{ a \} \cup \{ b \} = \{ a, b \} \not \in \rho$, so the second condition for $\rho$ to be a topological space is not satisfied. The collection $\rho$ is depicted below: