Approach Space - Examples

Examples

Every xpq-metric space (X,d) can be distancized to (X,d), as described at the beginning of the definition.

Given a set X, the discrete distance is given by d(x,A) = 0 if x ∈ A and = ∞ if x ∉ A. The induced topology is the discrete topology.

Given a set X, the indiscrete distance is given by d(x,A) = 0 if A is non-empty, and = ∞ if A is empty. The induced topology is the indiscrete topology.

Given a topological space X, a topological distance is given by d(x,A) = 0 if x ∈ A, and = ∞ if not. The induced topology is the original topology. In fact, the only two-valued distances are the topological distances.

Let P=, the extended positive reals. Let d+(x,A) = max (x−sup A,0) for x∈P and A⊆P. Given any approach space (X,d), the maps (for each A⊆X) d(.,A) : (X,d) → (P,d+) are contractions.

On P, let e(x,A) = inf { |x−a| : a∈A } for x<∞, let e(∞,A) = 0 if A is unbounded, and let e(∞,A) = ∞ if A is bounded. Then (P,e) is an approach space. Topologically, P is the one-point compactification of [0,∞). Note that e extends the ordinary Euclidean distance. This cannot be done with the ordinary Euclidean metric.

Let βN be the Stone–Čech compactification of the integers. A point U∈βN is an ultrafilter on N. A subset A⊆βN induces a filter F(A)=∩{U:U∈A}. Let b(U,A) = sup { inf { |n-j| : n∈X, j∈E } : X∈U, E∈F(A) }. Then (βN,b) is an approach space that extends the ordinary Euclidean distance on N. In contrast, βN is not metrizable.