---- > [!definition] Definition. ([[cover]]) > Let $X$ be a set. A collection $\mathscr{A}$ of subsets of $X$ whose union equals $X$ is called a **cover** of $X$. If $X$ is a [[topological space]], and $\mathscr{A}$ consists of open subsets of $X$, then $\mathscr{A}$ is called an **open covering** of $X$. > \ > Let $Y \subset X$. A collection $\mathscr{A}$ of subsets of $X$ is said to **cover the subset $Y$** if the union of its elements contains $Y$. > \ > Given a cover $\mathscr{A}$ of $X$, another cover $\mathscr{B}$ is called a **refinement of $\mathscr{A}$** if each $B \in \mathscr{B}$ is contained in some element $A \in \mathscr{A}$. It is an **open refinement** if each $B \in \mathscr{B}$ is an open subset of $X$. > [!basicexample] > - Every subcover of $\mathscr{A}$ is a refinement of $\mathscr{A}$. In general though, a refinement is probably not a subcover — it can contain elements that weren't originally in $\mathscr{A}$. ^basic-example > [!basicnonexample] Warning. > The intuition regarding the term 'refinement' here should be in terms a more 'refined' — parsimonious, removal of unneeded elements, preferable upon hindsight — choice of cover when compared to the original choice. In other situations, like [[comparable topologies|comparing topologies]], the term 'refinement' is interpreted as a more granular, fine-grained choice of set collection. Such intuitions somewhat contradict — for example, given topologies $\tau, \tau'$ with $\tau \subset \tau'$, we simultaneously have that $\tau'$ is *[[comparable topologies|finer]]* than $\tau$, while $\tau$ is a *refinement* of $\tau'$. Oops. ^nonexample ---- #### ---- #### 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 > ```