Notes on Quotients

Definition:

An equivalence relation on a set is a binary relation , denoted by the symbol "", such that for all

1. (Reflexivity)

2. (Symmetry) if then

3. (Transitivity) if and then

Examples of equivalence relations:

• 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."

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Definition:

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 .

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Definition:

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.

Examples: