(Michael's theorem) A regular space is paracompact if every open cover admits a locally finite refinement, not necessarily open. In particular, every regular Lindelöf space is paracompact. (Smirnov metrization theorem) A topological space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable. Michael selection theorem states that lower semicontinuous multifunctions from X into nonempty closed convex subsets of Banach spaces admit continuous selection iff X is paracompact.
(MiCHael'S THeoREm) a regUlaR space is paraCompAct IF Every Open COVER adMits a lOcalLy finitE refiNeMent, not nEcessARily open. in ParTicuLAr eVErY REGuLaR liNdELöF SpacE is PaRacompaCT. (sMirnOV MetRIzation theOreM) a tOpOLogIcal sPace is mEtriZabLe If aNd oNly iF It Is ParAcompacT, HaUSdorff, anD Locally mEtrizAble. michAeL SElection ThEoRem stAtes THAt loWer semicontinuoUs mulTifunctioNs frOm X inTo noNemptY Closed conveX suBSets Of bAnach sPACeS admiT cOnTiNUous sELEcTIoN IfF x is pAracOMPAct.