Set coverage

Coverage in mathematics is a family of sets, such that their union contains a given set.

Usually, coatings are considered in the general topology , where open coverings — families of open sets — are of most interest. In combinatorial geometry, an important role is played by coverings by convex sets ^[1] .

Content

1 Definitions
2 Related Definitions
3 Properties
4 See also
5 notes

Definitions

Let many be given $X$ ${\ displaystyle X}$ $X$ . Family of sets $C=\{U_{\alpha }\}_{\alpha \in A}$ ${\ displaystyle C = \ {U _ {\ alpha} \} _ {\ alpha \ in A}}$ $C=\{U_{{\alpha }}\}_{{\alpha \in A}}$ called coating $X$ ${\ displaystyle X}$ $X$ , if

X\subset \bigcup \limits _{\alpha \in A}U_{\alpha }.

{\ displaystyle X \ subset \ bigcup \ limits _ {\ alpha \ in A} U _ {\ alpha}.}

X\subset \bigcup \limits _{{\alpha \in A}}U_{{\alpha }}.

Let a topological space be given $(X,{\mathcal {T}})$ ${\ displaystyle (X, {\ mathcal {T}})}$ $(X,{\mathcal {T}})$ where $X$ ${\ displaystyle X}$ $X$ Is an arbitrary set, and ${\mathcal {T}}$ ${\ displaystyle {\ mathcal {T}}}$ $\mathcal{T}$ - defined on $X$ ${\ displaystyle X}$ $X$ topology . Then the family of open sets $C=\{U_{\alpha }\}_{\alpha \in A}\subset {\mathcal {T}}$ ${\ displaystyle C = \ {U _ {\ alpha} \} _ {\ alpha \ in A} \ subset {\ mathcal {T}}}$ $C=\{U_{{\alpha }}\}_{{\alpha \in A}}\subset {\mathcal {T}}$ called open cover set $Y\subset X$ ${\ displaystyle Y \ subset X}$ $Y\subset X$ , if

Y\subset \bigcup \limits _{\alpha \in A}U_{\alpha }.

{\ displaystyle Y \ subset \ bigcup \ limits _ {\ alpha \ in A} U _ {\ alpha}.}

Y\subset \bigcup \limits _{{\alpha \in A}}U_{{\alpha }}.

Related Definitions

If $C$ ${\ displaystyle C}$ $C$ - covering many $Y$ ${\ displaystyle Y}$ $Y$ then any subset $D\subset C$ ${\ displaystyle D \ subset C}$ $D\subset C$ also coating $Y$ ${\ displaystyle Y}$ $Y$ is called a subcover .
If each element of one coating is a subset of any element of the second coating, then they say that the first coating is inscribed in the second. More precisely, the coating $D=\{V_{\beta }\}_{\beta \in B}$ ${\ displaystyle D = \ {V _ {\ beta} \} _ {\ beta \ in B}}$ $D=\{V_{{\beta }}\}_{{\beta \in B}}$ inscribed on the cover $C=\{U_{\alpha }\}_{\alpha \in A}$ ${\ displaystyle C = \ {U _ {\ alpha} \} _ {\ alpha \ in A}}$ $C=\{U_{{\alpha }}\}_{{\alpha \in A}}$ , if

\forall \beta \in B\;\exists \alpha \in A

{\ displaystyle \ forall \ beta \ in B \; \ exists \ alpha \ in A}

\forall \beta \in B\;\exists \alpha \in A

such that

V_{\beta }\subset U_{\alpha }.

{\ displaystyle V _ {\ beta} \ subset U _ {\ alpha}.}

V_{{\beta }}\subset U_{{\alpha }}.

Coating $C=\{U_{\alpha }\}_{\alpha \in A}$ ${\ displaystyle C = \ {U _ {\ alpha} \} _ {\ alpha \ in A}}$ $C=\{U_{{\alpha }}\}_{{\alpha \in A}}$ many $Y$ ${\ displaystyle Y}$ $Y$ is called locally finite if for each point $y\in Y$ ${\ displaystyle y \ in Y}$ $y\in Y$ there is a neighborhood $U\ni y$ ${\ displaystyle U \ ni y}$ $U\ni y$ intersecting only with a finite number of elements $C$ ${\ displaystyle C}$ $C$ that is, many $\{\alpha \in A\mid U_{\alpha }\cap U\not =\varnothing \}$ ${\ displaystyle \ {\ alpha \ in A \ mid U _ {\ alpha} \ cap U \ not = \ varnothing \}}$ $\{\alpha \in A\mid U_{{\alpha }}\cap U\not =\varnothing \}$ of course .
Coating $C=\{U_{\alpha }\}_{\alpha \in A}$ ${\ displaystyle C = \ {U _ {\ alpha} \} _ {\ alpha \ in A}}$ $C=\{U_{{\alpha }}\}_{{\alpha \in A}}$ many $Y$ ${\ displaystyle Y}$ $Y$ called fundamental if any set whose intersection with each set $U\in C$ ${\ displaystyle U \ in C}$ $U\in C$ open in $Y$ ${\ displaystyle Y}$ $Y$ , itself open.
$Y$ ${\ displaystyle Y}$ $Y$ is called compact if any of its open coverings contains a finite subcovering;
$Y$ ${\ displaystyle Y}$ $Y$ is called paracompact if, in any of its open coverings, a locally finite open cover can be entered.

Properties

Any subcover is inscribed in the original coverage. The converse is generally not true.

Notes

↑ Coverage of the multitude - an article from the Mathematical Encyclopedia . A.V. Arkhangelsky, P.S. Soltan

[1] Coverage of the multitude - an article from the Mathematical Encyclopedia . A.V. Arkhangelsky, P.S. Soltan