An equivalence relation on a set
is a binary relation , denoted by the symbol
"
",
such that for all
(Reflexivity)
(Symmetry) if
then
(Transitivity) if
and
then
On
,
let
if
Let
and
be any to sets. Let
be any function. For
let
if
Note that any equivalence relation can be considered as generated by a function onto the set of "equivalences classes."
Let
be a family of topological spaces indexed
by
.
We consider the unlying sets to be "implicitly" disjoint, in particular we
consider
for
(
Let
to be the set of pairs
)
is defined to be the disjoint union of the underlying sets. The topology on
is defined as the finest topology on X for which all injections
are continuous.
is
open if and only if
is open for all
.
Given a topological space
,
a point set
,
and a function
which
is onto, we define a topology on
, called the quotient topology under
,
by defining
to
be open in the quotient topology if
is open in
.
We can also consider the quotient topology to be the "largest" topology (most
open sets) for which
is
continuous.