Contents

# 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.

notion of dimension

Last revised on March 21, 2021 at 03:30:24. See the history of this page for a list of all contributions to it.