6. Among all the subsets of the set $M=\{1,2,3, \cdots, 10\}$, there is a family of non-empty subsets such that the intersection of any two is not empty. What is the maximum number of such subsets?
(A) $2^{10}$ 
(B) $2^{9}$ 
(C) $10^{2}$ 
(D) $9^{2}$
6. (B).

Obviously, this family of subsets does not contain the empty set. According to the number of elements they contain, this family of subsets can be divided into 10 categories. Let's assume $A_{n_{i}}(i=1,2, \cdots, 10)$ is a subset containing $i$ elements, then the total number of subsets in this family that satisfy the given conditions is $\left(n_{1}+n_{2}+\cdots+n_{10}\right)$.

Obviously, the one-element subset $A_{n_{1}}$ is unique (otherwise, their intersection would all be the empty set). It is easy to know that $A_{n_{1}} \subseteq A_{n_{i}}(i=1,2, \cdots, 10)$, so $A_{n_{i}}$ can have at most $C_{9}^{i-1}$, i.e., $n_{i} \leqslant C_{9}^{i-1}$.
$$
\therefore n_{1}+n_{2}+\cdots+n_{10} \leqslant C_{9}^{0}+C_{9}^{1}+C_{9}^{2}+\cdots+C_{9}^{9}=2^{9} \text {. }
$$