nLab
characteristic function

Contents

Definition

Of a subset

The characteristic function of a subset UU of some set XX is a function from XX to the set TVTV of truth values (which classically is TV={,}TV = \{\bot,\top\}) that takes aa in XX to the truth value of the statement that aUa \in U. That is,

χ U(a)aU, \chi_U(a) \;\Leftrightarrow\; a \in U ,

where χ U\chi_U (also often 1 U1_U) is the characteristic function of UU.

Of a subobject

More generally, the characteristic morphism of a subobject UU of some objects XX in a category with a subobject classifier Ω\Omega is the morphism from XX to Ω\Omega that classifies UU; we have that

U X χ U 1 Ω \array { U & \hookrightarrow & X \\ \downarrow & & \downarrow & \chi_U \\ 1 & \underset{\top}\to & \Omega }

is a pullback square.

Revised on December 5, 2011 18:57:34 by Urs Schreiber (89.204.139.149)