---- > [!definition] Definition. ([[paracompact]]) > A [[topological space]] $X$ is called **paracompact** if every [[cover|open covering]] of $X$ admits a [[locally finite]] [[cover|open refinement]]. ^definition > [!equivalence] > If $X$ is also [[Hausdorff space|Hausdorff]], then "$X$ is paracompact" is equivalent to "$X$ admits a [[partition of unity]] subordinate to every [[cover|open cover]]". > ^equivalence > [!justification] Motivation. > This is a generalization of [[compact|compactness]], relaxing its requirements in two ways. >1. In the case of compactness, we require a *specific kind of open refinement* — one which is contained in the original open cover itself, (i.e., a subcover). >2. In the case of compactness, we enforce wholesale that said open refinement has finite [[cardinality]]. This condition is replaced with the notion of *locally* finite in the case of paracompactness. ^justification ---- #### ---- #### 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 > ```