Properties:: [[comparing topologies on R]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: [[basis-nestling characterization of comparing topologies]] ---- Let $X$ be a set. > [!definition] Definition. ([[comparable topologies]]) > Suppose that $\tau$ and $\tau'$ are two [[topological space|topologies]] on $X$. If $\tau' \supset \tau$, we say $\tau'$ is **finer** than $\tau$ and $\tau$ is **coarser** than $\tau$. In this case the two [[topological space|topologies]] are said to be **comparable**. > \ > When the inclusion is proper the terms **strictly finer** and **strictly coarser** may be used. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```