SomenewweakKnaster-Kuratouski-Mazurkiewicz(KKM)theoremsareprovedunderthenoncompactsituationinthegeneralizedfinitelycontinuousspace(GFCspace)withoutanyconvexity.Asapplications,theminimaxinequalitiesoftheKyFantypearealsogivenundersomesuitableconditions.Theresultsunifyandgeneralizesomeknownresultsinrecentliteratures.