Abstract: According to Urysohn's metrization theorem, every second axiom T3-space is metrizable. That is a compact Hausdorff space is metrizable if it has a countable base. This result is sometimes also called Urysohn's second metrization theorem. A family of subsets of a topological space is called locally finite. The countable union of locally finite (discrete) families is called σ-locally finite. Every discrete family is locally finite but not conversely. In a compact space, every locally finite or discrete family of subsets is finite. For every open covering of a metric space, there is a locally finite open cover that refines it. Every regular Lindelöf space is paracompact and every regular space that is either second axiom or σ-compact is paracompact. However, there are many unsolved problems concerning paracompact spaces. For example, it is not known whether the product of a paracompact space with the closed unit interval is even normal.
Publication Year: 1964
Publication Date: 1964-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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