The Quotient Topology

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Section The Quotient Topology

Given a topological space \(X\) and a surjection \(p\) from \(X\) to a set \(Y\text{,}\) we can use the topology on \(X\) to define a topology on \(Y\text{.}\) This topology on \(Y\) identifies points in \(X\) through the function \(p\text{.}\) The resulting function on \(Y\) is called a quotient topology. The quotient topology gives us a way of creating a topological space which models gluing and collapsing parts of a topological space.

Preview Activity 16.1.

Let \(X = \{1,2,3,4,5,6\}\) and let \(\tau = \{\emptyset, \{1,2\},\{4,6\}, \{1,2,4,6\},X\}\text{.}\) Let \(Y = \{a,b,c,d\}\) and define \(p: X \to Y\) by

\begin{equation*} p(1) = b, \ p(2) = a, \ p(3) = c, \ p(4) = d, \ p(5) = c, \ \text{ and } \ p(6) = a\text{.} \end{equation*}

Our goal in this activity is to define a topology on \(Y\) that is related to the topology on \(X\) via \(p\text{.}\)

(a)

We know the sets in \(X\) that are open. So let us consider the sets \(U\) in \(Y\) such that \(p^{-1}(U)\) is open in \(X\text{.}\) Define \(\sigma\) to be this set. That is

\begin{equation*} \sigma = \{U \subseteq Y \mid p^{-1}(U) \in \tau\}\text{.} \end{equation*}

Find all of the sets in \(\sigma\text{.}\)

(b)

Show that \(\sigma\) is a topology on \(Y\text{.}\)

(c)

Explain why \(p : (X, \tau) \to (Y, \sigma)\) is continuous.

(d)

Show that \(\sigma\) is the largest topology on \(Y\) for which \(p\) is continuous. That is, if \(\sigma'\) is a topology on \(Y\) with \(\sigma \subsetneqq \sigma'\text{,}\) then \(p: (X,\tau) \to (Y, \sigma')\) is not continuous.

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