TITLE: Meaning of n-connected pairs
QUESTION [8 upvotes]: A topological space $X$ is $n$-connected if the homotopy groups $\pi_r(X)$ for $0 \leq r \leq n$ are trivial groups. This means (let's say geometrically), $X$ is $0$-connected if it is non-empty and path-connected. It is $1$-connected if it is simply connected. 
Now, Let X be a topological space and A⊂X a subspace. For x∈A, the homotopy group $\pi_n(X,A,x)$ of the pair $(X,A)$ is by definition the set of homotopy classes of n-cells relative to X modulo A (the usual definition). Now, a pair is $n$-connected if $\pi_r(X,A)$ is trivial for $0 \leq r \leq n$. 
Is there any interpretation (geometrical) of $0$-connectedness and $1$-connectedness of a pair $(X,A)$? In general what does it mean (geometrically) by an $n$-connected pair?

REPLY [5 votes]: Here are two facts:

Any $n$-connected pair $(X,A)$ can be approximated by CW pairs $(X',A')\to (X,A)$ inducing isomorphisms on all the relative and absolute homotopy groups. 
A CW pair $(X',A')$ is $n$-connected iff up to homotopy equivalence of the pair, the cells of $X'-A'$ have dimensions $>n$. 

Intuitively, $n$-connected pairs $(X,A)$, up to weak homotopy equivlance (or homotopy equivalence if restricted to CW pairs), are such that $A$ "contains all dimension $\le n$ parts" of $X$.