CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A filtered space $X_*$ is called a connected filtered space if it satisfies:
$(\phi)_0$: The function $\pi_0X_0 \to \pi_0X_r$ induced by inclusion is surjective for all $r \gt 0$; and,
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:
$(\phi_0')$: 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,
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 $1 \leq j \leq i \lt r$.