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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!

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3  
I'd call it the (equivalence) relation that identifies the elements of $X$. –  joriki Nov 12 '11 at 13:39
3  
And I’d call $Y/\sim$ the quotient of $Y$ obtained by identifying/collapsing $X$ to a point. –  Brian M. Scott Nov 12 '11 at 13:50
    
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. –  Andreas Blass Nov 11 '12 at 2:31
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