nLab
dimension of a cell complex
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Contents
Idea
The dimension of a cell complex $X$ is the largest natural number $n \in \mathbb{N}$ , if such exists, for which there are non-trivial $n$ -cells in $X$ . If there is no such largest $n$ the dimension is also said to be infinite .

Definition
Specifically, if $X$ is a CW-complex

$X \;\simeq\; \underset{\longrightarrow_{\mathrlap{n}}}{\lim} X_n$

where each $X_n$ is a pushout of the form

$\array{
\underset{ i \in I_n }{\sqcup} S^{n-1}
&\longrightarrow&
X_{n-1}
\\
\big\downarrow &(po)& \big\downarrow
\\
\underset{ i \in I_n }{\sqcup} D^{n}
&\longrightarrow&
X_n
}$

the dimension of $X$ is the largest $n$ for which the indexing sets $I_n$ are non-empty .

Last revised on March 21, 2021 at 03:30:24.
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