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
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