1
$\begingroup$

Given any sets $X \subseteq Y$, the relation given by:

$1_Y \; \cup \; (X \times X) \;\; \subseteq Y \times Y$

(where $1_Y = \{ (y, y): y \in Y \}$ ) is an equivalence relation. Is there a name for this equivalence relation, and/or for the quotient of $Y\,$ induced by it?

(If I denote this quotient by $\frac{Y}{X}$, and the projection of $Y$ onto $\frac{Y}{X}$ by $p_X$, then $p_X$ sends all of $X$ to a single point, and maps $Y \;\; \backslash X$ bijectively.)

In particular, if $f$ is a Set morphism (i.e. function) with codomain $Y$, and $\mathrm{im}(f) \subseteq Y$ denotes the image of $f$, is there a name for the equivalence relation given by

$1_Y \; \cup \; (\mathrm{im}(f) \times \mathrm{im}(f)) \;\; \subseteq Y \times Y$

and/or for the quotient of $Y\,$ induced by it?

Thanks!

  • 0
    In topology one sometimes sees the notation $Y/X$ for this quotient. I haven't seen it (and don't recommend it) in set theory.2012-11-11

0 Answers 0