# nLab connected filtered space

### Context

#### Topology

topology

algebraic topology

## Definition

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

1. $(\phi)_0$: The function $\pi_0X_0 \to \pi_0X_r$ induced by inclusion is surjective for all $r \gt 0$; and,

2. for all $i \geq 1$, $(\phi_i): \pi_i(X_r,X_i,v)=0$ for all $r \gt i$ and $v \in X_0$.

Another equivalent form is:

1. $(\phi_0')$: The function $\pi_0X_s \to \pi_0X_r$ induced by inclusion is surjective for all $0=s \lt r$ and bijective for all $1 \leq s \leq r$; and,

2. for all $i \geq 1$, $(\phi_i'): \, \pi_j(X_r,X_i,v)=0$ for all $v \in X_0$ and all $j,r$ such that $1 \leq j \leq i \lt r$.

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