Section 20: The Metric Topology
is a metric on if is a non-negative symmetric function such that iff , and the triangle inequality holds. is called the distance between and .
is a metric space if is a metric on and the topology on [called the metric topology induced by ] is generated by the basis consisting of -balls centered at , , for all and . A topological space is called metrizable if there is a metric on that induces .
General properties
- The topology induced by is the coarsest topology on such that is continuous.
- The standard bounded metric corresponding to is . and induce the same topology.
- Another example of a bounded metric inducing the same topology as is .
Standard metrics on
- isthe euclidean metric on if where .
- isthe square metric on if .
- isthe uniform metric on if .
- The uniform metric inducesthe uniform topology.
- For , where . is open in the box topology but not in the uniform topology.
- is a metric that induces the product topology on .
Properties
- The euclidean and square metrics induce the standard topology on .
- And so does every metric of the form for .
- The box topology on is finer than the uniform topology which is finer than the product topology. If is infinite, all three are different.
- , for all , which is a homeomorphism relative to the box and product topologies, is continuous relative to the uniform topology iff s are bounded from above, and is a homeomorphism iff s are bounded from below and above by some positive numbers.
Metrization of
[in the box or product topology] is metrizable only if is countable and the topology is the product topology.
- The metric is one that induces the product [box and uniform] topology on .
- The metric is one that induces the product topology on .
- As we shall see in §21, if and is metrizable, then there is a sequence of elements of converging to .
- in the box topology is not metrizable. If then in the box topology, but there is clearly no sequence of elements of converging to in the box topology. A similar argument works for all infinite .
- in the product topology is not metrizable if is uncountable. If , then in the product topology, but, if is uncountable, then for any sequence of points of there is some such that for all , hence, does not converge to in the product topology.
Subspaces of
is the subset of consisting of all sequences such that converges. Then, the topology induced by on is called the -topology.
is the subset of consisting of all sequences that are eventually zero.
- equals in the box topology, the set of all sequences of real numbers converging to in the uniform topology, and in the product topology.
- The -topology is strictly finer than the uniform topology on , but strictly coarser than the box topology [both inherited from ].
- When all four [box, , uniform, and product] topologies are inherited by , they are all distinct.
- When all four [box, , uniform, and product] topologies are inherited by the Hilbert cube, , the box topology is strictly finer than the other three topologies which become equal.