nLab
connected filtered space

Definition

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

1. $(\varphi {)}_0$: The function ${\pi }_0{X}_0\to {\pi }_0{X}_r$ induced by inclusion is surjective for all $r>0$; and,

2. for all $i\ge 1$, $({\varphi }_i):{\pi }_i(X_r,X_i,v)=0$ for all $r>i$ and $v\in X_0$.

Another equivalent form is:

1. $({\varphi }_0\prime )$: The function ${\pi }_0{X}_s\to {\pi }_0{X}_r$ induced by inclusion is surjective for all $0=s<r$ and bijective for all $1\le s\le r$; and,

2. for all $i\ge 1$, $({\varphi }_i\prime ):{\pi }_j(X_r,X_i,v)=0$ for all $v\in X_0$ and all $j,r$ such that $1\le j\le i<r$.