Comparable Topologies on a Set

Comparable Topologies on a Set

For an arbitrary set $X$ there are many different topologies that can be obtained. Sometimes these topologies are comparable in the sense that one of the topologies is contained in the other.

Definition: Let $X$ be a set and $\tau_1$ and $\tau_2$ be two topologies defined on $X$. If either $\tau_1 \subseteq \tau_2$ or $\tau_1 \supseteq \tau_2$ then $\tau_1$ and $\tau_2$ are said to be Comparable. If $\tau_1 \subseteq \tau_2$ then $\tau_2$ is said to be a Finer topology than $\tau_1$ (or Strictly Finer if $\tau_1 \neq \tau_2$) and $\tau_1$ is said to be Coarser topology than $\tau_2$ (or Strictly Coarser of $\tau_1 \neq \tau_2$).

For example, consider the set $X = \{ a, b, c, d, e \}$ and the nested topology $\tau_1 = \{ \emptyset, \{a \}, \{a, b \}, \{a, b, c \}, \{a, b, c, d \}, X \}$.

Now consider the following topology:

(1)
\begin{align} \quad \tau_2 = \{ \emptyset, \{a \}, \{b \}, \{a, b \}, \{a, b, c \}, \{a, b, c, d \}, X \} \end{align}

So $\tau_1 \subset \tau_2$, so the finer topology is $\tau_1$ and the coarser topology is $\tau_2$. Now consider the this topology:

(2)
\begin{align} \quad \tau_3 = \{ \emptyset, \{ a \}, \{b \}, \{ c \} \{a , b \}, \{a, c \}, \{ b, c \} \{a, b, c \}, \{a, b, c , d \}, X \} \end{align}

So $\tau_1 \subset \tau_2 \subset \tau_3$. Between these three topologies, $\tau_1$ is the coarsest and $\tau_3$ is the finest.

For another example, consider the set of real numbers $\mathbb{R}$ with the usual topology $\tau_1$ of open intervals. Recall from The Lower and Upper Limit Topologies on the Real Numbers page that the lower limit topology is the topology, denote it $\tau_2$, generated by all unions of elements in $\{ [a, b) : a, b \in \mathbb{R}, a \leq b \}$ and the upper limit topology is the topology, denote it $\tau_3$, generated by all unions of elements in $\{ (a, b] : a, b \in \mathbb{R}, a, b \}$.

Notice that all open intervals can be written as the union of open sets in the lower limit topology and in the upper limit topology since for the interval $(c, d)$ we have that:

(3)
\begin{align} \quad (c, d) = \bigcup_{c < a} [a, d) \in \tau_2 \quad \mathrm{and} \quad (c, d) = \bigcup_{b < d} (c, b] \in \tau_3 \end{align}

Therefore $\tau_1 \subset \tau_2$ and $\tau_1 \subset \tau_3$. Hence, the lower and upper limit topologies on $\mathbb{R}$ are both finer than the usual topology on $\mathbb{R}$.

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