covering

Etymology 1
Present participle or participial adjective from ; or, from participle form of,.

Etymology 2
From, , , [ verbal noun of ,  ] ; or, verbal noun from.

Noun

 * 1)  That which covers or conceals; a cover; something spread or laid over or wrapped about another.
 * 2)  Action of the verb to cover.
 * 3)  A collection of sets $$\{U_i\}$$ such that their union contains $$U$$; such a collection with the additional property that every $$U_i$$ is open; such a collection with the additional property that every $$U_i$$ is contained in $$U$$.
 * 4)  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 $$ \pi : E \rightarrow X$$ between topological spaces such that there exists, for every point $$x$$ in $$X$$, a discrete space $$D_x$$ and an open neighborhood $$U$$ of $$x$$ such that $$\pi^{-1}(U)= \displaystyle \bigsqcup_{d \in D_x} V_d $$ and $$\pi|_{V_d}:V_d \rightarrow U $$ is a homeomorphism for every $$d \in D_x $$. See
 * 1)  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 $$ \pi : E \rightarrow X$$ between topological spaces such that there exists, for every point $$x$$ in $$X$$, a discrete space $$D_x$$ and an open neighborhood $$U$$ of $$x$$ such that $$\pi^{-1}(U)= \displaystyle \bigsqcup_{d \in D_x} V_d $$ and $$\pi|_{V_d}:V_d \rightarrow U $$ is a homeomorphism for every $$d \in D_x $$. See

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

Translations

 * Armenian:, ,
 * Bulgarian: ,
 * Finnish:
 * German:
 * Gothic: 𐌷𐌿𐌻𐌹𐍃𐍄𐍂
 * Greek:
 * Ancient: σκέπασμα
 * Hebrew: כיסוי, עֲטִיפָה
 * Hungarian: ,
 * Ingrian: kate
 * Irish: cumhdach
 * Latin: integumentum, operculum, tegimen, tegumentum
 * Manx: coodagh
 * Maori: tūpuni
 * Scottish Gaelic: sgàile, pùball