Section 22*: The Quotient Topology

• A surjective $p$ is a quotient map iff ($V$ is closed in $Y$ iff $p^{-1}(V)$ is closed in $X$ ).
• $p$ is a quotient map iff it is surjective, continuous and maps open saturated sets to open sets, where $A$ in $X$ is called saturated if it is the preimage of some set in $Y$ .
• A quotient map does not have to be open or closed, a quotient map that is open does not have to be closed and vice versa.

Properties of quotient maps

• A restriction of a quotient map to a subdomain may not be a quotient map even if it is still surjective (and continuous).
  • If $p:X\rightarrow Y$ is a quotient map and $A\subset X$ is saturated with respect to $p$ , then if $A$ is open or closed, or $p$ is open or closed, then $p|_{A}:A\rightarrow p(A)$ is a quotient map.
  • If $p:X\rightarrow Y$ is an open quotient map and $A\subset X$ is open, then $p|_{A}:A\rightarrow p(A)$ is an open quotient map.
• The composite of two quotient maps is a quotient map.
• The product of two quotient maps may not be a quotient map.
  • If both quotient maps are open then the product is an open quotient map.
  • Another condition guaranteeing that the product is a quotient map is the local compactness (see Section 29).

Showing that a function is a quotient map

• If a continuous function has a continuous right inverse then it is a quotient map.
• A retraction is a quotient map. A retraction of $X$ onto $A\subset X$ is a continuous map $f:X\rightarrow A$ such that for $x\in A$ : $f(x)=x$ .

Quotient topology and quotient space

• Let $X^{*}$ be a partition of the space $X$ with the quotient topology induced by $p:X\rightarrow X^{*}$ where $p(x)=A$ such that $x\in A$ , then $X^{*}$ is called a quotient space of $X$ .
  • One can think of the quotient space as a formal way of "gluing" different sets of points of the space.
• For $X^{*}$ to satisfy the $T_{1}$ -axiom we need all sets in $X^{*}$ to be closed.
  • For $X^{*}$ to be a Hausdorff space there are more complicated conditions.

Quotient spaces and continuous functions