Covering

English

Pronunciation

• Template:IPA
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Etymology 1

Present participle or participial adjective from Template:Affix; or, from Template:Inh participle form of Template:M, Template:M.

Verb

1. Template:Infl of

Etymology 2

From Template:Inh, Template:M, Template:M, Template:M [verbal noun of Template:M, Template:M]; or, verbal noun from Template:Affix.

Noun

1. Template:Lb That which covers or conceals; a cover; something spread or laid over or wrapped about another.
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2. Template:Lb Action of the verb to cover.
3. Template:Lb 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. Template:SenseidTemplate:Lb 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 $\pi :E\to X$ 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 ${\pi }^{-1}(U)=\underset{d\in D_x}{⨆}{V}_d$ and $\pi {|}_{V_d}:V_d\to U$ is a homeomorphism for every $d\in D_x$. See Template:Pedia

Usage notes

In sense Template:Senseno, 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

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