# Topology/Quotient Spaces

The quotient topology is not a natural generalization of anything studied in analysis, however it is easy enough to motivate. One motivation comes from geometry. For example, the torus can be constructed by taking a rectangle and pasting the edges together.

## Definition: Quotient Map[edit]

Let and be topological spaces; let be a surjective map. The map * f * is said to be a * quotient map * provided a is open in * Y * if and only if is open in * X *.

## Definition: Quotient Map Alternative[edit]

There is another way of describing a quotient map. A subset is * saturated * (with respect to the surjective map ) if * C * contains every set that it intersects. To say that * f * is a quotient map is equivalent to saying that * f * is continuous and * f * maps saturated open sets of * X * to open sets of * Y *. Likewise with closed sets.

There are two special types of quotient maps: * open maps * and * closed maps *.

A map is said to be an * open map if for each open set , the set is open in Y . A map is said to be a closed map if for each closed , the set is closed in Y . It follows from the definition that if is a surjective continous map that is either open or closed, then f is a quotient map.
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## Definition: Quotient Topology[edit]

If * X * is a topological space and * A * is a set and if is a surjective map, then there exist exactly one topology on * A * relative to which * f * is a quotient map; it is called the * quotient topology * induced by * f *.

## Definition: Quotient Space[edit]

Let * X * be a topological space and let , be a partiton of * X * into disjoint subsets whose union is * X *. Let be the surjective map that carries each to the element of containing it. In the quotient topology induced by * f * the space is called a * quotient space * of * X *.

## Theorem[edit]

Let be a quotient map; let * A * be a subspace of * X * that is saturated with respect to * f *; let be the map obtained by restricting * f *, then * g * is a quotient map.

1.) If * A * is either opened or closed in * X *.

2.) If * f * is either an open map or closed map.

Proof: We need to show:

when V

and

when .

Since and * A * is saturated, . It follows that both and equal all points in * A * that are mapped by * f * into * V *. For the second equation, for any two subsets * U * and

In the opposite direction, suppose when and . Since * A * is saturated, , so that in particular . Then where .

* Suppose A or f * is open. Since , assume is open in and show

*V*is open in .

First, suppose * A * is open. Since is open in * A * and * A * is open in * X *, is open in * X *. Since , is open in * X *. * V * is open in * Y * because * f * is a quotient map.

Now suppose * f * is open. Since and is open in A, for a set * U * open in * X *. Now because * f * is surjective; then

The set is open in * Y * because * f * is an open map; hence * V * is open in . The proof for closed * A * or * f * is left to the reader.