Editing Metrizable space (section)

==Examples==

The group of unitary operators <math>\mathbb{U}(\mathcal{H})</math> on a separable Hilbert space <math>\mathcal{H}</math> endowed
with the [[strong operator topology]] is metrizable (see Proposition II.1 in <ref>Neeb, Karl-Hermann, On a theorem of S. Banach. J. Lie Theory 7 (1997), no. 2, 293–300.</ref>).

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 [[Function (mathematics)|function]]s from the [[real line]] <math>\R</math> to itself, with the [[topology of pointwise convergence]].

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.

===Locally metrizable but not metrizable===

The [[Line with two origins]], also called the ''{{dfn|bug-eyed line}}'' is a [[non-Hausdorff manifold]] (and thus cannot be metrizable). Like all manifolds, it is [[locally homeomorphic]] to [[Euclidean space]] and thus [[locally metrizable space|locally metrizable]] (but not metrizable) and [[Locally Hausdorff space|locally Hausdorff]] (but not [[Hausdorff space|Hausdorff]]). It is also a [[T1 space|T<sub>1</sub>]] [[locally regular space]] but not a [[semiregular space]].

The [[Long line (topology)|long line]] is locally metrizable but not metrizable; in a sense, it is "too long".