Definitions
1.Verbpresent participle and gerund of cover
2.NounThat which covers or conceals; a cover; something spread or laid over or wrapped about another.
3.NounAction of the verb to cover.
4.NounA collection of sets
{
U
i
}
{\displaystyle \{U_{i}\}}
such that their union contains
U
{\displaystyle U}
; such a collection with the additional property that every
U
i
{\displaystyle U_{i}}
is open; such a collection with the additional property that every
U
i
{\displaystyle U_{i}}
is contained in
U
{\displaystyle U}
.
5.NounA 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
π
:
E
→
X
{\displaystyle \pi :E\rightarrow X}
between topological spaces such that there exists, for every point
x
{\displaystyle x}
in
X
{\displaystyle X}
, a discrete space
D
x
{\displaystyle D_{x}}
and an open neighborhood
U
{\displaystyle U}
of
x
{\displaystyle x}
such that
π
−
1
(
U
)
=
⨆
d
∈
D
x
V
d
{\displaystyle \pi ^{-1}(U)=\displaystyle \bigsqcup _{d\in D_{x}}V_{d}}
and
π
|
V
d
:
V
d
→
U
{\displaystyle \pi |_{V_{d}}:V_{d}\rightarrow U}
is a homeomorphism for every
d
∈
D
x
{\displaystyle d\in D_{x}}
. See Covering space on Wikipedia.Wikipedia