Strong F Compact Sets In L-topological Space, Ultra-f Compact Set, And The Local Super-f1 Compactness,

Compactness alway play an ential roe in topology. Untill now,there have been seveal definitions of fuzzy compactness on fuzzy topological spaces,among which are N - compact, compactness of Chang, strong fuzzy compactnessand ultra - fuzzy compactness, ect. N - compactness has been used by many schol-ars for its nice properties. What we really concerned is how about the last two fuzzycompactnesses. Strong fuzzy compactness and ultra - fuzzy compactness were de-fined for fuzzy topological spaces by Lowen, and generalized to L - fuzzy topologi-cal spaces (where L is a completely distributive complete lattice with an order - re-versing involution ') by Wang and Kudri. The present paper will extend strongfuzzy compactness and ultra- fuzzy compactness to an arbitrary L - set for an arbi-trary complete lattice L. The results we obtained verifies the importance of thesetwo fuzzy compactness. It also lays a basis for studying local ultra - F₁ compactnessof L- topological spaces which is the third chapter of this paper. The definition ofthe local ultra - F₁ compactness can generalize some important theorems and propo-sitions which are related to local compactness in general topology to fuzzy topologyunder the same assumptions or stronger assumptions. There are three chapters inthis paper:Preparation knowledge, in chapter one, defines some notions and gives somelemmas and theorems we will use in the paper. For example, weakly induced, co-prime element, L - valued Zadeh function,α- remote neighborhood family (α -RF for shOrt) of L-- set A and Pnduct sPace ect.in chapter tWo, fbotly, we define strOng F2 comactness and ultra-- F2 cotn-pactness of an L-- set A by conSdering the comactness of its all cut ':P(A) = lxl A(x) 7 a I (a e M(L)). Strong F2 comact L-- sets and uhra-- F2 cornactL-- sets, just like N-- Wact L-- sets, are PrOVed to hav som USeftil pIDPerheseven under the W assWtion that L is a comlete lattice. bondly, we dealwith stng Fl Hactness and u1tra -- F1 COInactness of an L -- set A by studyngthe comactness of all sets c.'(A) = 1x l A(x) $a' I (a e M' .7xA(x))),and me results correSpoding Ston F2 comact L -- sets and ultra -- F2 COInact L-- sets are established. At las, the relations between theSe defined tw comact L-- sets (induding N -- comact L -- sets) are investigated.In chaPter three, the concePt of locally ultra -- Fl comact L -- topobeicalSPace is introduced. We shOw that local ultra -- F1 comactness is multipilcative andWtary for both H L -- setS and cM L -- sets. We obtain fuzZy VerSbo ofsom other classical pMties Of ha Hactness. W, we investwite ultra-- Fl Wactness and local ultra -- F1 comactness of induced I(L ) -- hay topo-logied space. It is obtained that (I(L )x, w(8)) is a haly tiltra -- Fl comactsPace if f (Lx, 8) is a locally ultra -- F1 comact sPace.