Equivalent metrics and compactifications

Let (X, d) be a metric space and m ∈ X. Suppose that φ : X×X → R is a nonnegative symmetric function. We define a metric d on X which is equivalent to d. If d is totally bounded, its completion is a compactification of (X, d). As examples, we construct two compactifications of (R, dE), where dE is the Euclidean metric and s ≥ 2. key words. equivalent metric; completion; compactification Mathematics Subject Classifications (2000). 54E35, 54D35 1 The metric d Let (X, d) be a metric space and m ∈ X . Suppose that φ : X × X → R is a nonnegative symmetric function. As usual, two metrics d1 and d2 on a set X are called equivalent if (X, d1) and (X, d2) are homeomorphic. In this section, we will define a metric d on X which is equivalent to d. For each x, y ∈ X , let δ(x, y) = min { d(x, y), 1 1 + d(m,x) + φ(x, y) + 1 1 + d(m, y) } . And for each x, y ∈ X and n ∈ N, let Γnx,y = { (x0, · · · , xn) | x0 = x, xn = y and xi ∈ X for all i } and Γx,y = ⋃ n∈N Γnx,y. Notice that Γx,y 6= ∅ for all x, y ∈ X . In the following definition, the infimum runs over all elements of Γx,y.