Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B) ≠ ∅, prove that A∩∂ (B) ≠ ∅. Where ∂ (B) is the boundary of B.

a) Determine if the vectors are linearly dependent or independent. b) Is it possible for the set (V1, v2,03, V4} to be considered as a base of R4 c) Consider the set H = Gen{V,02,03, vA) Is it a linear space? If so, how is its dimension found?

Let X=ℝ2 and define d2,:ℝ2×ℝ2→ℝ by d2((x1 ,y1),(x2,y2)) = max{|x1 - x2|,|y1 - y2|}. a) Verify that d2 is a metric on ℝ2. b.) Draw the neighborhood N((0; 1) for d2, where 0 is the origin in ℝ2.