Lebesgue's number lemma

In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states: In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states: Such a number δ {displaystyle delta } is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well. Let U {displaystyle {mathcal {U}}} be an open cover of X {displaystyle X} . Since X {displaystyle X} is compact we can extract a finite subcover { A 1 , … , A n } ⊆ U {displaystyle {A_{1},dots ,A_{n}}subseteq {mathcal {U}}} .If any one of the A i {displaystyle A_{i}} 's equals X {displaystyle X} then any δ > 0 {displaystyle delta >0} will serve as a Lebesgue number.Otherwise for each i ∈ { 1 , … , n } {displaystyle iin {1,dots ,n}} , let C i := X ∖ A i {displaystyle C_{i}:=Xsetminus A_{i}} , note that C i {displaystyle C_{i}} is not empty, and define a function f : X → R {displaystyle f:X ightarrow mathbb {R} } by f ( x ) := 1 n ∑ i = 1 n d ( x , C i ) {displaystyle f(x):={frac {1}{n}}sum _{i=1}^{n}d(x,C_{i})} . Since f {displaystyle f} is continuous on a compact set, it attains a minimum δ {displaystyle delta } . The key observation is that δ > 0 {displaystyle delta >0} .If Y {displaystyle Y} is a subset of X {displaystyle X} of diameter less than δ {displaystyle delta } , then there exist x 0 ∈ X {displaystyle x_{0}in X} such that Y ⊆ B δ ( x 0 ) {displaystyle Ysubseteq B_{delta }(x_{0})} , where B δ ( x 0 ) {displaystyle B_{delta }(x_{0})} denotes the ball of radius δ {displaystyle delta } centered at x 0 {displaystyle x_{0}} (namely, one can choose as x 0 {displaystyle x_{0}} any point in Y {displaystyle Y} ). Since f ( x 0 ) ≥ δ {displaystyle f(x_{0})geq delta } there must exist at least one i {displaystyle i} such that d ( x 0 , C i ) ≥ δ {displaystyle d(x_{0},C_{i})geq delta } . But this means that B δ ( x 0 ) ⊆ A i {displaystyle B_{delta }(x_{0})subseteq A_{i}} and so, in particular, Y ⊆ A i {displaystyle Ysubseteq A_{i}} . Munkres, James R. (1974), Topology: A first course, p. 179, ISBN 978-0-13-925495-6.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:''''''''''''}.mw-parser-output .citation .cs1-lock-free a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url('//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png')no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}