808: n-Cube Centered at p with Edges-Length l with Indexes B

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definition of $n$-cube centered at $p$ with edges-length $l$ with indexes $B$

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of subspace topology of subset of topological space.
• The reader knows a definition of Euclidean topological space.

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Target Context

• The reader will have a definition of $n$-cube centered at $p$ with edges-length $l$ with indexes $B$.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
${\mathbb{R}}^d$: $=\text{ the Euclidean topological space }$
$p$: $\in {\mathbb{R}}^d$, $=(p^1,...,p^d)$
$l$: $\in \mathbb{R}$ such that $0\le l$
$n$: $\in \mathbb{N}$ such that $0\le n\le d$
$B$: $\subseteq \{1,...,d\}$ such that $|B|=n$
$\ast C_{p,l,B}$: $=\{p^{\prime }\in {\mathbb{R}}^d|\mathrm{\forall }j\in \{1,...,d\}\setminus B(p^{\prime j}=p^j)\wedge \mathrm{\forall }j\in B(p^j-l/2\le p^{\prime j}\le p^j+l/2)\}$ with the subspace topology
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Conditions:
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2: Note

Of course, this definition is not particularly interesting, but we have this definition in order to avoid explaining every time what we mean by "n-cube centered at $p$ with edges-length $l$ with indexes B".

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References

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