Pointed space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a pointed space or based space is a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as 
 
 
 
 
 x
 
 0
 
 
 ,
 
 
 {\displaystyle x_{0},}
 
 that remains unchanged during subsequent discussion, and is kept track of during all operations.
Maps of pointed spaces (based maps) are <a href="/facts/Continuous_(topology)/sKbl02pB">continuous maps</a> preserving basepoints, i.e., a map 
 
 
 
 f
 
 
 {\displaystyle f}
 
 between a pointed space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with basepoint 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 and a pointed space 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 with basepoint 
 
 
 
 
 y
 
 0
 
 
 
 
 {\displaystyle y_{0}}
 
 is a based map if it is continuous with respect to the topologies of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 and if 
 
 
 
 f
 
 (
 
 x
 
 0
 
 
 )
 
 =
 
 y
 
 0
 
 
 .
 
 
 {\displaystyle f\left(x_{0}\right)=y_{0}.}
 
 This is usually denoted

f
        :
        
          (
          
            X
            ,
            
              x
              
                0
              
            
          
          )
        
        →
        
          (
          
            Y
            ,
            
              y
              
                0
              
            
          
          )
        
        .
      
    
    {\displaystyle f:\left(X,x_{0}\right)\to \left(Y,y_{0}\right).}

Pointed spaces are important in <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, particularly in <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a>, where many constructions, such as the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a>, depend on a choice of basepoint.
The <a href="/facts/Pointed_set/EELNN0pq">pointed set</a> concept is less important; it is anyway the case of a pointed <a href="/facts/Discrete_space/hpvXA23u">discrete space</a>.
Pointed spaces are often taken as a special case of the <a href="/facts/Relative_topology/NjU6z4Ca">relative topology</a>, where the subset is a single point. Thus, much of <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a> is usually developed on pointed spaces, and then moved to relative topologies in <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>.

Pointed space open-in-new

Pointed space