J. Munkres. Topology (2013) Chapter 2: Topological Spaces and Continuous Functions. 22: The Quotient Topology

Definition

Let \(X\) and \(Y\) be topological spaces. Let \(p : X \rightarrow Y\). Then \(p\) is strongly continuous iff, for all \(U \subseteq Y\), we have \(U\) is open in \(Y\) if and only if \(p^{-1}(U)\) is open in \(X\).

Definition

Let \(X\) and \(Y\) be topological spaces. Let \(p : X \rightarrow Y\). Then \(p\) is a quotient map iff \(p\) is surjective and strongly continuous.

Definition

Let \(X\) and \(Y\) be sets. Let \(p : X \rightarrow Y\). Let \(C \subseteq X\). Then \(C\) is saturated iff, for all \(x,y \in X\), if \(x \in C\) and \(p(x) = p(y)\) then \(y \in C\).

Proposition

Let \(X\) and \(Y\) be topological spaces. Let \(p : X \rightarrow Y\) be surjective. Then the following are equivalent.

  1. \(p\) is a quotient map.
  2. \(p\) is continuous and maps saturated open sets to open sets.
  3. \(p\) is continuous and maps saturated closed sets to closed sets.

Definition

Let \(X\) and \(Y\) be topological spaces. Let \(f : X \rightarrow Y\). Then \(f\) is a closed map iff it maps closed sets to closed sets.

Proposition

Every continuous open map is a quotient map. Every continuous closed map is a quotient map.

Example

\(\pi_1 : \{ (x,y) \in \mathbb{R}^2 \mid x \geq 0 \text{ or } y = 0 \} \rightarrow \mathbb{R}\) is a quotient map that is neither an open map nor a closed map.

Example 1

Define \(p : [0,1] \cup [2,3] \rightarrow [0,2]\) by \(p(x) = x\) if \(x \in [0,1]\) and \(p(x) = x-1\) if \(x \in [2,3]\). Then \(p\) is a quotient map and a closed map, but not an open map.

Example 2

The projection \(\pi_1 : \mathbb{R}^2 \rightarrow \mathbb{R}\) is a quotient map and an open map, but not a closed map.

Definition

Let \(X\) be a topological space, \(A\) a set, and \(p : X \rightarrow A\) a surjection. Then the quotient topology on \(A\) is the unique topology such that \(p\) is a quotient map, namely \(\{U \in \mathcal{P} A \mid p^{-1}(U) \text{ is open in } X\}\).

Definition

Let \(X\) be a topological space. Let \(X^*\) be a partition of \(X\). Let \(\pi : X \rightarrow X^*\) be the canonical projection. Then \(X^*\) under the quotient topology is called a quotient space of \(X\).

The following is the first difficult theorem in Munkres. At the moment (paraphrasing Thorsten Altenkirch) I understand each step of the proof, but I do not understand the proof.

Theorem 22.1

Let \(p : X \rightarrow Y\) be a quotient map. Let \(A\) be a subspace of \(X\) that is saturated with respect to \(p\). Let \(q : A \rightarrow p(A)\) be the function obtained by restricting \(p\).

  1. If \(A\) is either open or closed in \(X\), then \(q\) is a quotient map.
  2. If \(p\) is either an open map or a closed map, then \(q\) is a quotient map.

Proof

  • \(\langle 1 \rangle 1\) For all \(V \subseteq p(A)\), we have \(q^{-1}(V) = p^{-1}(V)\).
    • Let \(V \subseteq p(A)\).
    • \(q^{-1}(V) \subseteq p^{-1}(V)\)
      • Set theory.
    • \(p^{-1}(V) \subseteq q^{-1}(V)\)
      • Let \(x \in p^{-1}(V)\)
      • \(p(x) \in V\)
      • \(p(x) \in p(A)\)
      • \(x \in A\)
        • \(A\) is saturated.
      • \(q(x) = p(x) \in V\)
      • \(x \in q^{-1}(V)\)
  • \( \langle 1 \rangle 2 \) For all \(U \subseteq X\), we have \(p(U \cap A) = p(U) \cap p(A)\).
    • Let \(U \subseteq X\).
    • \(p(U \cap A) \subseteq p(U) \cap p(A)\)
      • Set theory.
    • \(p(U) \cap p(A) \subseteq p(U \cap A)\)
      • Let \(y \in p(U) \cap p(A)\)
      • Pick \(u \in U\) and \(a \in A\) such that \(p(u) = p(a) = y\)
      • \(u \in A\)
        • \(A\) is saturated.
      • \(u \in U \cap A\)
      • \(y \in p(U \cap A)\)
  • If \(A\) is open then \(q\) is a quotient map.
    • Assume \(A\) is open.
    • For any \(V \subseteq p(A)\), if \(q^{-1}(V)\) is open in \(A\) then \(V\) is open in \(p(A)\).
      • Let \(V \subseteq p(A)\).
      • Assume \(q^{-1}(V)\) is open in \(A\).
      • \(q^{-1}(V)\) is open in \(X\).
        • Lemma 16.2
      • \(p^{-1}(V)\) is open in \(X\).
        • By \(\langle 1 \rangle 1 \).
      • \(V\) is open in \(Y\).
        • \(p\) is a quotient map.
      • \(V\) is open in \(p(A)\).
        • \(V = V \cap p(A)\)
    • \(q\) is a quotient map.
  • If \(p\) is an open map then \(q\) is a quotient map.
    • \( \langle 2 \rangle 1 \) Assume \(p\) is an open map.
    • For any \(V \subseteq p(A)\), if \(q^{-1}(V)\) is open in \(A\) then \(V\) is open in \(p(A)\).
      • Let \(V \subseteq p(A)\).
      • Assume \(q^{-1}(V)\) is open in \(A\).
      • \(p^{-1}(V)\) is open in \(A\).
        • \(\langle 1 \rangle 1 \)
      • \( \langle 3 \rangle 1 \) Pick \(U\) open in \(X\) such that \(p^{-1}(V) = U \cap A\).
      • \( \langle 3 \rangle 2 \) \(p(p^{-1}(V)) = V\)
        • Because \(p\) is surjective.
      • \(V = p(U) \cap p(A)\)
        • \[ \begin{align} V & = p(p^{-1}(V)) & (\langle 3 \rangle 2) \\ & = p(U \cap A) & (\langle 3 \rangle 1) \\ & = p(U) \cap p(A) & (\langle 1 \rangle 2) \end{align} \]
      • \(p(U)\) is open in \(Y\).
        • \( \langle 2 \rangle 1 \)
      • \(V\) is open in \(p(A)\).
    • \(q\) is a quotient map.
  • If \(A\) is closed then \(q\) is a quotient map.
    • Assume \(A\) is closed.
    • For any \(V \subseteq p(A)\), if \(q^{-1}(V)\) is closed in \(A\) then \(V\) is closed in \(p(A)\).
      • Let \(V \subseteq p(A)\).
      • Assume \(q^{-1}(V)\) is closed in \(A\).
      • \(q^{-1}(V)\) is closed in \(X\).
        • Theorem 17.3
      • \(p^{-1}(V)\) is closed in \(X\).
        • By \(\langle 1 \rangle 1 \).
      • \(V\) is closed in \(Y\).
        • \(p\) is a quotient map.
      • \(V\) is closed in \(p(A)\).
        • \(V = V \cap p(A)\)
    • \(q\) is a quotient map.
  • If \(p\) is a closed map then \(q\) is a quotient map.
    • \( \langle 2 \rangle 1 \) Assume \(p\) is a closed map.
    • For any \(V \subseteq p(A)\), if \(q^{-1}(V)\) is closed in \(A\) then \(V\) is closed in \(p(A)\).
      • Let \(V \subseteq p(A)\).
      • Assume \(q^{-1}(V)\) is closed in \(A\).
      • \(p^{-1}(V)\) is closed in \(A\).
        • \(\langle 1 \rangle 1 \)
      • \( \langle 3 \rangle 1 \) Pick \(U\) closed in \(X\) such that \(p^{-1}(V) = U \cap A\).
        • Theorem 17.2
      • \( \langle 3 \rangle 2 \) \(p(p^{-1}(V)) = V\)
        • Because \(p\) is surjective.
      • \(V = p(U) \cap p(A)\)
        • \[ \begin{align} V & = p(p^{-1}(V)) & (\langle 3 \rangle 2) \\ & = p(U \cap A) & (\langle 3 \rangle 1) \\ & = p(U) \cap p(A) & (\langle 1 \rangle 2) \end{align} \]
      • \(p(U)\) is closed in \(Y\).
        • \( \langle 2 \rangle 1 \)
      • \(V\) is open in \(p(A)\).
    • \(q\) is a quotient map.
  • \(\Box\)

Proposition

The composite of two quotient maps is a quotient map.

Example 7

The product of two quotient maps is not necessarily a quotient map. Let \(X = \mathbb{R}\) and let \(X^*\) be the quotient space formed by identifying all the points in \(\mathbb{Z}_+\). Let \(p : X \rightarrow X^*\) be the quotient map. We show that \(p \times \mathrm{id}_\mathbb{Q} : X \times \mathbb{Q} \rightarrow X^* \times \mathbb{Q}\) is not a quotient map.

  • For \(n \in \mathbb{Z}_+\), let \(c_n = \sqrt{2} / n\).
  • For \(n \in \mathbb{Z}_+\), let
    • \[ U_n = \{(x,y) \in \mathbb{R} \times \mathbb{Q} \mid n-1/4 < x < n + 1/4, (y < c_n + n - x \wedge y < c_n - n + x) \vee (y > c_n + n - x \wedge y > c_n - n + x)\} \]
  • For all \(n \in \mathbb{Z}_+\), \(U_n\) is open in \(\mathbb{R} \times \mathbb{Q}\)
  • For all \(n \in \mathbb{Z}_+\), \(\{n\} \times \mathbb{Q} \subseteq U_n\)
  • Let \(U = \bigcup_{n \in \mathbb{Z}_+} U_n\)
  • \(U\) is saturated with respect to \(p \times \mathrm{id}_{\mathbb{Q}}\)
    • Since \((n,q) \in U\) for all \(n \in \mathbb{Z}_+\) and \(q \in \mathbb{Q}\).
  • \((p \times \mathrm{id}_\mathbb{Q})(U)\) is not open in \(X^* \times \mathbb{Q}\)
    • Let \(U' = (p \times \mathrm{id}_\mathbb{Q})(U)\)
    • Let \(b = p(n)\) for all \(n \in \mathbb{Z}_+\).
    • \((b,0) \in U'\)
    • Pick neighbourhoods \(W\) of \(b\) and \(\delta > 0\) such that \(W \times ((-\delta,\delta) \cap \mathbb{Q}) \subseteq U'\)
    • \(p^{-1}(W) \times ((-\delta, \delta) \cap \mathbb{Q}) \subseteq U\)
    • Pick \(n \in \mathbb{Z}_+\) such that \(c_n < \delta\)
    • Pick \(0 < \epsilon < 1/4\) such that \((n-\epsilon, n+\epsilon) \subseteq p^{-1}(W)\)
    • Pick a rational number \(q\) such that \(|q - c_n| < \epsilon / 2\)
    • \((n + \epsilon / 2, q) \in U\)
    • This is a contradiction.
  • \(\Box\)

Theorem 22.2

Let \(p : X \rightarrow Y\) be strongly cont. Let \(Z\) be a topological space. Let \(f : Y \rightarrow Z\).

  1. If \(f \circ p\) is continuous then \(f\) is continuous.
  2. If \(f \circ p\) is a quotient map then \(f\) is a quotient map.

From the Exercises

Definition

Let \(X\) be a topological space and \(A \subseteq X\). A retraction of \(X\) onto \(A\) is a continuous map \(r : X \rightarrow A\) such that \(r(a) = a\) for all \(a \in A\).

Proposition

Every retraction is a quotient map.

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