This is the closure relation in Quantum Mechanics:
$$\sum_i |i\rangle \langle i| = 1 $$
which I understand as "the sum of the projections onto the basis vectors leaves the projected vector unchanged" (as long as $\lbrace |i \rangle \rbrace$ is a basis).
In the Quantum Mechanics book I am reading it is stated:
Given two subspaces $V_i$ and $V_j$, we define their sum as $V_i \oplus V_j$ as the set containing (1) all elements of $V_i$, (2) all elements of $V_j$, (3) all combinations of the above. But for the elements (3), closure would be lost.
What is the meaning of the sentence in bold type?