Examples of Non-metrizable Spaces
Non-normal spaces cannot be metrizable; important examples include
- the Zariski topology on an algebraic variety or on the spectrum of a ring, used in algebraic geometry,
- the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence.
- the Strong operator topology on the set of unitary operators on a Hilbert Space (often denoted )
The real line with the lower limit topology is not metrizable. The usual distance function is not a metric on this space because the topology it determines is the usual topology, not the lower limit topology. This space is Hausdorff, paracompact and first countable.
The long line is locally metrizable but not metrizable; in a sense it is "too long".
Read more about this topic: Metrization Theorem
Famous quotes containing the words examples of, examples and/or spaces:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)