# nLab connected filtered space

### Context

#### Topology

topology

algebraic topology

## Definition

A filtered space ${X}_{*}$ is called a connected filtered space if it satisfies:

1. $\left(\varphi {\right)}_{0}$: The function ${\pi }_{0}{X}_{0}\to {\pi }_{0}{X}_{r}$ induced by inclusion is surjective for all $r>0$; and,

2. for all $i\ge 1$, $\left({\varphi }_{i}\right):{\pi }_{i}\left({X}_{r},{X}_{i},v\right)=0$ for all $r>i$ and $v\in {X}_{0}$.

Another equivalent form is:

1. $\left({\varphi }_{0}\prime \right)$: The function ${\pi }_{0}{X}_{s}\to {\pi }_{0}{X}_{r}$ induced by inclusion is surjective for all $0=s and bijective for all $1\le s\le r$; and,

2. for all $i\ge 1$, $\left({\varphi }_{i}\prime \right):\phantom{\rule{thinmathspace}{0ex}}{\pi }_{j}\left({X}_{r},{X}_{i},v\right)=0$ for all $v\in {X}_{0}$ and all $j,r$ such that $1\le j\le i.

Revised on October 26, 2010 13:09:31 by Urs Schreiber (87.212.203.135)