Metric Spaces, Generalized Euclidean Space

Generalized Euclidean Space

Take the direct sum of arbitrarily many copies of the real line, using zero as the preferred element. As you recall, every point in this composite space has finitely many nonzero coordinates; all the other coordinates are 0.

Call this space E_j, for Jeneralized euclidean Space. By default, E_j joins a countable collection of real lines together, but higher cardinalities are possible. Of course, if you join a finite number of real lines together you simply get Rⁿ, i.e. n dimensional space.

Construct a metric on E_j as follows. Take any two points and note that their coordinates agree almost everywhere. In fact their coordinates are 0 almost everywhere. Subtract the coordinates that disagree, square the differences, add them up, and take the square root. Since two points can only disagree on a finite number of coordinates, this is well defined.

Take any three points and derive their distances, using the above formula. Note that all three points live in Rⁿ for some integer n. Furthermore, our distance metric is identical to the metric on n space, Therefore the triangular inequality holds. The metric is valid across all of E_j.

Use open balls to build a base for the topology of E_j as usual.

Wait a minute! The direct sum of spaces already has its own topology. Does this metric give the same topology?

Let Q be a base open set in E_j, according to the direct sum topology. Thus Q restricts a finite set of coordinates to various open intervals, and leaves the other coordinates unconstrained. Let p be a point in this open set, and let p act as the center of an open ball, using our metric. If Q constrains the i^th coordinate, and p_i is a distance r_i from the nearer end of the open interval, let the radius of our ball be anything less than r_i. Since there are finitely many intervals, the radius is well defined. The resulting ball lies entirely inside Q, and Q is covered by open balls. The metric produces a topology that is at least as strong as the "direct sum" topology.

Next suppose some base set Q fits into the ball of radius 1. Let z be any point in Q and change one of its unconstrained coordinates from 0 to 2. This is still a point in Q, still in the direct sum, yet it is not contained in our open ball. Thus E_j has open sets that are not present in the traditional direct sum. Its topology is stronger.