In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a nested sequence of compact subsets 
  
    
      
        
          K
          
            i
          
        
      
    
    {\displaystyle K_{i}}
  
 of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 (i.e. 
  
    
      
        
          K
          
            1
          
        
        ⊆
        
          K
          
            2
          
        
        ⊆
        
          K
          
            3
          
        
        ⊆
        ⋯
      
    
    {\displaystyle K_{1}\subseteq K_{2}\subseteq K_{3}\subseteq \cdots }
  
), such that 
  
    
      
        
          K
          
            i
          
        
      
    
    {\displaystyle K_{i}}
  
 is contained in the interior of 
  
    
      
        
          K
          
            i
            +
            1
          
        
      
    
    {\displaystyle K_{i+1}}
  
, i.e. 
  
    
      
        
          K
          
            i
          
        
        ⊆
        
          int
        
        (
        
          K
          
            i
            +
            1
          
        
        )
      
    
    {\displaystyle K_{i}\subseteq {\text{int}}(K_{i+1})}
  
  for each 
  
    
      
        i
      
    
    {\displaystyle i}
  
 and 
  
    
      
        X
        =
        
          ⋃
          
            i
            =
            1
          
          
            ∞
          
        
        
          K
          
            i
          
        
      
    
    {\displaystyle X=\bigcup _{i=1}^{\infty }K_{i}}
  
.  A space admitting an exhaustion by compact sets is called exhaustible by compact sets.
 For example, consider 
  
    
      
        X
        =
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle X=\mathbb {R} ^{n}}
  
 and the sequence of closed balls 
  
    
      
        
          K
          
            i
          
        
        =
        {
        x
        :
        
          |
        
        x
        
          |
        
        ≤
        i
        }
        .
      
    
    {\displaystyle K_{i}=\{x:|x|\leq i\}.}
  

 Occasionally some authors drop the requirement that 
  
    
      
        
          K
          
            i
          
        
      
    
    {\displaystyle K_{i}}
  
 is in the interior of 
  
    
      
        
          K
          
            i
            +
            1
          
        
      
    
    {\displaystyle K_{i+1}}
  
, but then the property becomes the same as the space being σ-compact, namely a countable union of compact subsets.
 The following are equivalent for a topological space 
  
    
      
        X
      
    
    {\displaystyle X}
  
:
 (where weakly locally compact means locally compact in the weak sense that each point has a compact neighborhood).
 The hemicompact property is intermediate between exhaustible by compact sets and σ-compact.  Every space exhaustible by compact sets is hemicompact and every hemicompact space is σ-compact, but the reverse implications do not hold.  For example, the Arens-Fort space and the Appert space are hemicompact, but not exhaustible by compact sets (because not weakly locally compact), and the set 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  
 of rational numbers with the usual topology is σ-compact, but not hemicompact.
 Every regular space exhaustible by compact sets is paracompact.