Topology/Quotient Spaces

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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.

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 partition 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  


  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.

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