Metric Spaces, Alternate Bases

Alternate Bases

Within the xy plane, the euclidean distance function acts as a metric, and defines a metric space. The base for the topology consists of open disks of all radii and centers. However, many other bases are possible. For instance, one can use squares instead of circles. Build a base using the interior of squares of all sizes, and at all locations throughout the plane. Given an open disk, and a point p inside that disk, p is always a certain distance from the edge, and can be enclosed in one of our squares, which is wholly contained in the disk. Thus the disk is an open set, and similarly for all other disks. That reproduces our original topology.

Generalizing to higher dimensions, a topology based on blocks, i.e. hypercubes, is the same as the topology based on open balls. A ball can always be covered with tiny blocks, and an open block can be covered with open balls.

In fact we can use any shape at all, as long as it is open and bounded. Draw the shape of a turtle in the plane, and build a base consisting of the interior of turtles, all sizes, all locations. Sure enough, each open disk can be covered by open turtles, and each turtle can be covered by open disks, so the topologies are the same. Besides, you can tell your friends, "It's turtles all the way down."