# 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 together 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.*

## 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.