Let $X$ be a topological space and $A$ a subspace of $X$. Given $k\geq 2$, let the unordered configuration space be $$ B(X,k)=\{(x_1,x_2,\cdots,x_k)\in X^k\mid x_i\neq x_j \text{ for any } i\neq j\} $$ and the relative unordered configuration space be $$ B(X,A;k)=\{(x_1,x_2,\cdots,x_k)\in X^k\mid x_i\neq x_j \text{ for any } i\neq j,\text{ and } x_t\in A\text{ for some } 1\leq t\leq k\}. $$

**Question:** I want to express $B(X,A;k)$ in terms of $B(Y,i)$ 's for some space $Y$.

I have tried: For $j=1,2,\cdots, k$, first choose $j$ points in $A$, then choose $k-j$ points in $X\setminus A$. Then we have $$ B(X,A;k)=\bigsqcup_{j=1}^k B(A,j)\times B(X\setminus A,k-j). $$ However, this expression seems wrong since it is a non-connected disjoint union... Where is the problem?