Examples of Non-normal Spaces
An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry.
A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. More generally, a theorem of A. H. Stone states that the product of uncountably many non-compact metric spaces is never normal.
Read more about this topic: Normal Space
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