How Do You Prove Metric Space? To verify that (S, d) is a metric space, we should first check that if d(x, y) = 0 then x = y. This follows from the fact that, if γ is a path from x to y, then L(γ) ≥ |x − y|, where |x − y| is
Is RA Metric Space? Theorem: R is a complete metric space — i.e., every Cauchy sequence of real numbers converges. Is QA metric space? A metric space is separable space if it has a countable dense subset. Typical examples are the real numbers or any Euclidean space. For metric spaces (but not for general topological
How Do You Show That A Metric Space Is Complete? A metric space (X, ρ) is said to be complete if every Cauchy sequence (xn) in (X, ρ) converges to a limit α ∈ X. There are incomplete metric spaces. If a metric space (X, ρ) is not complete then it has Cauchy sequences that
Is A Discrete Set Closed? Sometimes a discrete set is also closed. Then there cannot be any accumulation points of a discrete set. On a compact set such as the sphere, a closed discrete set must be finite because of this. Is discrete metric space open or closed? Any subset A can be written as