Topologies on Sets

Definition: Let $X$ be a set. A Topology on $X$ is a collection $\tau$ of subsets of $X$ that satisfies the following properties: 1) $X, \emptyset \in \tau$. 2) If $\{ U_i : i \in I \}$ is any arbitrary collection of subsets of $X$ such that $U_i \in \tau$ for all $i \in I$ then the union $\displaystyle{\bigcup_{i \in I} U_i \in \tau}$. 3)) If $\{ U_1, U_2, ..., U_n \}$ is any finite collection of subsets of $X$ such that $U_i \in \tau$ for all $i \in \{ 1, 2, ..., n \}$ then the intersection $\displaystyle{\bigcap_{i=1}^{n} U_i \in \tau}$. The pair, $(X, \tau)$ is called a Topological Space.

Note that (2) states that the union of any ARBITRARY collection of sets in $\tau$ is also contained in $\tau$, while (3) states that the intersection of any FINITE collection of sets in $\tau$ is also contained in $\tau$. In other words, a topological space $(X, \tau)$ is closed under arbitrary unions and under finite intersections.

It can easily be show that for any set $X$ that $\tau = \{ \emptyset, X \}$ and $\tau = \mathcal P (X)$ (the power set of $X$ which is the collection of all subsets of $X$) are both topologies on $X$. These topologies are given a special name.

Definition: The Trivial Topologies on $X$ are $\tau = \{ \emptyset, X \}$ and $\tau = \mathcal P (X)$.

We now give an example of a set $X$ and a nontrivial topology $\tau$ on $X$.

Consider the following set:

(1)

\begin{align} \quad X = \{ a, b, c, d, e, f \} \end{align}

And the following collection of subsets of $X$:

(2)

\begin{align} \quad \tau = \{ \emptyset, \{ a, b \} \{ c, d, e, f \}, X \} \end{align}

We now show $\tau$ is a topological space. We see that $\emptyset, X \in \tau$ so (1) is satisfied. We now list all of the unions of sets from $\tau$:

(3)

\begin{align} \quad \emptyset \cup \{ a, b \} &= \{ a, b \} \\ \quad \emptyset \cup \{ c, d, e, f \} &= \{ c, d, e, f \} \\ \quad \emptyset \cup X &= X \\ \quad \{a, b\} \cup \{ c, d, e, f \} &= X \\ \quad \{a, b \} \cup X &= X \\ \quad \{c, d, e, f \} \cup X &= X \\ \\ \quad \emptyset \cup \{ a, b \} \cup \{ c, d, e, f \} &= X \\ \quad \emptyset \cup \{ a, b \} \cup X &= X \\ \quad \emptyset \cup \{ c, d, e, f \} \cup X &= X \\ \quad \{a, b \} \cup \{ c, d, e, f \} \cup X &= X \\ \\ \quad \emptyset \cup \{ a, b \} \cup \{ c, d, e, f \} \cup X &= X \end{align}

Each of these arbitrary unions is contained in $\tau$ so (2) is satisfied. We now list all of the intersections of sets from $\tau$:

(4)

\begin{align} \quad \emptyset \cap \{ a, b \} &= \emptyset \\ \quad \emptyset \cap \{ c, d, e, f \} &= \emptyset \\ \quad \emptyset \cap X &= \emptyset \\ \quad \{a, b\} \cap \{ c, d, e, f \} &= \emptyset \\ \quad \{a, b \} \cap X &= \{ a, b \} \\ \quad \{c, d, e, f \} \cap X &= \{ c, d, e, f \} \\ \\ \quad \emptyset \cap \{ a, b \} \cap \{ c, d, e, f \} &= \emptyset \\ \quad \emptyset \cap \{ a, b \} \cap X &= \emptyset \\ \quad \emptyset \cap \{ c, d, e, f \} \cap X &= \emptyset \\ \quad \{a, b \} \cap \{ c, d, e, f \} \cap X &= \{ a, b \} \\ \\ \quad \emptyset \cap \{ a, b \} \cap \{ c, d, e, f \} \cap X &= \emptyset \end{align}

Each of these intersections (which are finite since $X$ is finite) are contained in $\tau$ so (3) is satisfied and hence $\tau$ is a topology on $X$.