Dense order

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Partial_order/qb4a3FAX">partial order</a> or <a href="/facts/Total_order/xnTtmuYJ">total order</a> < on a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is said to be dense if, for all 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and 
  
    
      
        y
      
    
    {\displaystyle y}
  
 in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 for which 
  
    
      
        x
        <
        y
      
    
    {\displaystyle x<y}
  
, there is a 
  
    
      
        z
      
    
    {\displaystyle z}
  
 in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 such that 
  
    
      
        x
        <
        z
        <
        y
      
    
    {\displaystyle x<z<y}
  
. That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are <a href="/facts/Comparability/PvLeJXol">comparable</a>.
</p>

Dense order open-in-new

Dense order