definition of \(n\)-cube centered at \(p\) with edges-length \(l\) with indexes \(B\)
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of subspace topology of subset of topological space.
- The reader knows a definition of Euclidean topological space.
Target Context
- The reader will have a definition of \(n\)-cube centered at \(p\) with edges-length \(l\) with indexes \(B\).
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.
Main Body
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 \(\vert B \vert = n\)
\(*C_{p, l, B}\): \(= \{p' \in \mathbb{R}^d \vert \forall j \in \{1, ..., d\} \setminus B (p'^j = p^j) \land \forall j \in B (p^j - l / 2 \le p'^j \le p^j + l / 2)\}\) 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".
