Proof of Existence of Lebesgue Number for Open Cover of a Compact Metric Space
Ifis a compact metric space andis an open cover ofthen there issuch thatfor some %alpha for any
That is, a numberexists such that the- neighbourhoodof any pointis a subset of at least one
The numberis called the Lebesgue number of the cover.
Sinceis a cover ofeachbelongs to at least one
Eachis open, hence for eachexists such thatfor some
The collection of setsis an open cover ofand sinceis compact, a finite subcoverexists.
The Lebesgue number is defined as