General Topology: Topological Spaces
Comparability
If $\mathcal{T}$ and $\mathcal{J}$ are two topologies on a set $X$, then $\mathcal{T}$ is finer than $\mathcal{J}$ if $\mathcal{J} \subseteq \mathcal{T}$, and $\mathcal{T}$ is strictly finer than $\mathcal{J}$ if $\mathcal{J} \subset \mathcal{T}$. Conversely, $\mathcal{T}$ is coarser than $\mathcal{J}$ if $\mathcal{T} \subseteq \mathcal{J}$, and $\mathcal{T}$ is strictly coarser than $\mathcal{J}$ if $\mathcal{T} \subset \mathcal{J}$. A finer topology is like a higher resolution screen and a coarser topology is like a lower resolution screen in that the higher resolution screen can display all the detail of the lower resolution screen plus more. Two topologies $\mathcal{T}$ and $\mathcal{J}$ are comparable if either $\mathcal{T} \subseteq \mathcal{J}$ or $\mathcal{J} \subseteq \mathcal{T}$.
Problems
Let $X = \{1, 2\}$. List all the possible topologies on $X$ and note which are finer than others and which ones are not comparable.
There are four possible topologies: $\{\varnothing, X\}$, $\{\varnothing, \{1\}, X\}$, $\{\varnothing, \{2\}, X\}$, and $\{\varnothing, \{1\}, \{2\}, X\}$. $\{\varnothing, \{1\}, \{2\}, X\}$ is finer than the other three. $\{\varnothing, \{1\}, X\}$ and $\{\varnothing, \{2\}, X\}$ are not comparable, but both are finer than $\{\varnothing, X\}$.