Let V be any normed linear space with norm ∥⋅∥. Given a set D⊆V and a set E⊆D, we say that E is relatively open in D, or sometimes that E is an open subset of D if there is an open set U in V such that E=U∩D. Note that this definition is Definition 9.2.2 in the book. In Theorem 9.2.3 in the book, an equivalent definition is given. In other words, the set V already has a topology, that is, a collection of subsets that are defined as open, and if we restrict our attention to a subset D of V (which may not be open), then we can define a topology on D as the collection of relatively open subsets of D. 4. Let E⊆D and suppose that for every x0​∈E there exists δ>0 such that B(x0​,δ)∩D⊆ E, where B(x0​,δ)={y∈V:∥y−x∥<δ} is the ball centered at x0​ of radius δ. (Note: this is the direction of the proof of Theorem 9.2 .3 not done in the book.)