Pointed set
In mathematics, a pointed set (also based set or rooted set) is an ordered pair ( X , x 0 ) {\displaystyle (X,x_{0})} where X {\displaystyle X} is a set and x 0 {\displaystyle x_{0}} is an element of X {\displaystyle X} called the base point (also spelled basepoint). A map between pointed a sets ( X , x 0 ) {\displaystyle (X,x_{0})} and ( Y , y 0 ) {\displaystyle (Y,y_{0})} —called a based map, pointed map, or point-preserving map—is a function from X {\displaystyle X} to Y {\displaystyle Y} that maps one basepoint to another, i.e.