Why Do We Need The Requirement of "Ordered by Inclusion" in this Theorem?

Question

My functional analysis textbook provides the following theorem:

"The union of a collection of subspaces totally ordered by inclusion is a subspace."

Why do we need the requirement of "totally ordered by inclusion" for this theorem to hold?

Answer 1

Consider a collection of two subspaces of $\mathbb R^2$ which is not totally ordered. Is the union a subspace?