covering

• IPA^(key): /ˈkʌvəɹɪŋ/

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Present participle or participial adjective from cover (verb) +‎ -ing; or, from Middle English participle form of coveren, keveren (“to cover”).

covering

1. present participle and gerund of cover

From Middle English coverynge, -inge, keverynge, -inge [verbal noun of coveren, keveren (“to cover”)];[1] or, verbal noun from cover (verb) +‎ -ing.

covering (countable and uncountable, plural coverings)

1. (countable) That which covers or conceals; a cover; something spread or laid over or wrapped about another.
   • 2020 May 20, Richard Clinnick, “Network News: More trains... but advice is not to use public transport”, in Rail, page 6:

     Face covering is also mentioned: "There are some circumstances when wearing a face covering may be marginally beneficial as a precautionary measure. The evidence suggests that wearing a face covering does not protect you, but it may protect others if you are infected but have not yet developed symptoms. [...] If you can, wear a face covering if you need to use public transport."

2. (uncountable) Action of the verb to cover.
3. (topology, of a given open set ${\displaystyle U}$) A collection of sets ${\displaystyle \{U_{i}\}}$ such that their union contains ${\displaystyle U}$; such a collection with the additional property that every ${\displaystyle U_{i}}$ is open; such a collection with the additional property that every ${\displaystyle U_{i}}$ is contained in ${\displaystyle U}$.
4. (topology) A special kind of map that establishes a relationship between two topological spaces in the sense that, under the action of the map, the one looks locally like several copies of the second: Formally, a continuous map ${\displaystyle \pi :E\rightarrow X}$ between topological spaces such that there exists, for every point ${\displaystyle x}$ in ${\displaystyle X}$, a discrete space ${\displaystyle D_{x}}$ and an open neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ such that ${\displaystyle \pi ^{-1}(U)=\displaystyle \bigsqcup _{d\in D_{x}}V_{d}}$ and ${\displaystyle \pi |_{V_{d}}:V_{d}\rightarrow U}$ is a homeomorphism for every ${\displaystyle d\in D_{x}}$. See Covering space on Wikipedia.Wikipedia

In sense sense 4, properly, the map ${\displaystyle \pi }$ is the covering, and ${\displaystyle E}$ is called the covering space, but by abuse of terminology, "covering" may refer to ${\displaystyle E}$, "covering space" may refer to ${\displaystyle \pi }$, and either may refer to the tuple ${\displaystyle (E,\pi )}$.