Suppose $k$ is fixed. Consider a set $X$ of subsets of the ground set $\{1, \dots, k \}$, with the following property: there is some ordering of the elements of $X$, as $X = \{ x_1, \dots, x_n \}$, such that the chain of sets $Y_j = \cup_{i=1}^j x_i$ is *strictly* increasing, i.e.
$$
\emptyset \subsetneq Y_1 \subsetneq Y_2 \dots \subsetneq Y_n
$$

Note that $X$ cannot contain the empty set. When $k = 2$, the set $X$ can be equal to $\{ \{1 \} \}$, $\{ \{2 \} \}$, $\{ \{1 \}, \{1, 2 \} \}$, $\{ \{2 \}, \{1, 2\} \}$, or $\{ \{1 \}, \{2 \} \}$, or $\{ \{1, 2 \} \}$. But $X$ is not allowed to be $\{ \{1 \}, \{2 \}, \{1, 2\} \}$.

Is there a simpler way of expressing this criterion for the set $X$? (That does not require a reference to an explicit ordering of its elements) Has this type of structure been studied before? For example, how many possibilities are there for $X$?

It appears based on some examples that $X$ can be written in this way iff the set $X$ is linearly independent over $GF(2)^k$ (with the obvious correspondence between subsets of $\{1, ... k \}$ and vectors over $GF(2)^k$. I have some trouble proving this criterion