Nested Topologies

Nested Topologies

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 at a type of topology known as a nested topology.

 Definition: Let $X \neq \emptyset$. A Nested Topology is a collection $\tau = \{ \emptyset, X \} \cup \{ U_1, U_2, ... \} \cup \left \{ \bigcup_{i=1}^{\infty} U_i \right \}$ of subsets of $X$ that satisfy $\emptyset \subset U_1 \subset U_2 ... \subset U_n \subset ...$.

Let's indeed verify that $(X, \tau)$ is a topological space. Let $X \neq \emptyset$ and let $\tau = \tau = \{ \emptyset, X \} \cup \{ U_1, U_2, ... \} \cup \left \{ \bigcup_{i=1}^{\infty} U_i \right \}$ where the subsets of $X$] in [[$\tau$ satisfy the nesting:

(1)
\begin{align} \quad \emptyset \subset U_1 \subset U_2 \subset ... \subset U_n \subset ... \quad (*) \end{align}

By definition we have that $\emptyset, X \in \tau$ so the first condition is satisfied.

For the second condition, let $I$ be an indexing set and consider the arbitrary collection $\{ U_i \}_{i \in I}$ of sets in $\tau$. If $\{ U_i \}_{i \in I}$ is a finite collection of subsets of $X$ from $\tau$ then there exists a set in $\{ U_i \}_{i \in I}$ with a largest index, say $n \in I$ is such that $n \geq i$ for all $i \in I$. Then:

(2)
\begin{align} \quad \bigcup_{i \in I} U_i = U_n \in \tau \end{align}

If $\{ U_i \}_{i \in I}$ is instead an infinite collection of subsets of $X$ from $\tau$, then since $U_i \in \tau$ for each $i \in I$ and we have the nesting from $(*)$, we see that $\bigcup_{i \in I} U_i = \bigcup_{i=1}^{\infty} U_i \in \tau$. In either case, we see that the second condition is satisfied.

For the third condition, let $U_{k_1}, U_{k_2}, ..., U_{k_n} \in \tau$ be any finite collection of subsets of $X$ from $\tau$ where $k_1 < k_2 < ... < k_n$. Then from $(*)$ we have that:

(3)
\begin{align} \quad U_{k_1} \subset U_{k_2} \subset ... \subset U_{k_n} \end{align}

The intersection of this finite nested collection of subsets of $X$ from $\tau$ is therefore:

(4)
\begin{align} \quad \bigcap_{i=1}^{n} U_{k_i} = U_{k_1} \in \tau \end{align}

Therefore the third condition is satisfied and $(X, \tau)$ is a topological space.