Quotient Spaces

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Now we examine how to define a topology on a quotient set. ([Set Theory] §Equivalence Relations, ⁋Definition 4)

Locally closed subspace

Definition 1 Let \(X\) be a topological space. A subset \(A\) is called locally closed at \(x\in A\) if there exists a neighborhood \(V\) of \(x\) in \(X\) such that \(A\cap V\) is closed in \(V\). If \(A\) is locally closed at every \(x\in A\), then we call \(A\) itself locally closed.

Proposition 2 For a topological space \(X\) and a subset \(A\), the following are all equivalent.

1. \(A\) is locally closed in \(X\).
2. \(A\) is the intersection of an open set and a closed set in \(X\).
3. \(A\) is open in its closure \(\cl A\) (in \(X\)).

Proof

First, suppose \(A\) is locally closed, and for each \(x\in A\) let \(V_x\) be an open neighborhood of \(x\) in \(X\) satisfying the condition of Definition 1. Then \(U=\bigcup_{x\in A} V_x\) is an open set. Also, by applying §Subspaces, ⁋Proposition 5, we see that \(A\) is closed in \(U\). Thus \(A=U\cap C\) for some closed set \(C\) in \(X\), so the second condition holds.

Now suppose \(A=U\cap C\) holds for an open set \(U\) and a closed set \(C\) in \(X\). Then \(\cl A\subseteq C\), so

\[A\subseteq U\cap\cl A\subseteq U\cap C=A\]

holds, and in particular \(A=U\cap\cl A\). From this we see that \(A\) is open in \(\cl A\).

Finally, if there exists an open set \(U\) in \(X\) satisfying \(A=U\cap\cl A\), then \(A\) is closed in \(U\) and hence locally closed.

In particular, from condition 2, if a continuous function \(f:X\rightarrow Y\) and a locally closed subset \(B\) of \(Y\) are given, then \(f^{-1}(B)\) is also locally closed in \(X\).

Quotient Spaces

Definition 3 Let a topological space \(X\) be given, and let \(R\) be an equivalence relation on the set \(X\). Then the quotient space of \(X\) by \(R\) means the space \(X/R\) equipped with the final topology defined by the canonical projection \(p:X\rightarrow X/R\).

By §Initial and Final Topology, ⁋Proposition 5, the open sets in \(X/R\) are exactly those sets \(U\) such that \(p^{-1}(U)\) is open in \(X\).¹ Rewriting this in the language of [Set Theory] §Examples of Equivalence Relations, ⁋Definition 5, we can verify that the open sets on \(X/R\) correspond bijectively to the open sets of \(X\) that are saturated with respect to \(R\).

Meanwhile, by §Initial and Final Topology, ⁋Proposition 6, the following holds.

Proposition 4 Let a topological space \(X\), a quotient space \(X/R\), and the canonical projection \(p:X\rightarrow X/R\) be given. For any topological space \(Y\), a function \(f:X/R\rightarrow Y\) is continuous if and only if \(f\circ p\) is a continuous function from \(X\) to \(Y\).

Proposition 5 Consider a topological space \(X\) and two equivalence relations \(R,S\) defined on \(X\). If \(S\) is a finer equivalence relation than \(R\), then the bijection \((X/S)/(R/S)\rightarrow X/R\) is a homeomorphism for the equivalence relation \(R/S\) defined on \(X/S\).

Proof

That \((X/S)/(R/S)\rightarrow X/R\) is a bijection was already shown in [Set Theory] §Examples of Equivalence Relations, ⁋Definition 8. By Proposition 4, the continuity of this function is equivalent to the continuity of \(X/S\rightarrow X/R\), and the continuity of this function in turn follows from the continuity of \(X\rightarrow X/R\).

Similarly, the continuity of \(X/R\rightarrow(X/S)/(R/S)\) is obtained from the continuity of \(X\rightarrow(X/S)/(R/S)\), and this function is the composition of two continuous functions

\[X\longrightarrow X/S\longrightarrow (X/S)/(R/S)\]

so it is continuous.

Meanwhile, let topological spaces \(X,Y\) and a continuous function \(f:X\rightarrow Y\) be given, and consider the equivalence relation \(R\) defined by \(f\). ([Set Theory] §Examples of Equivalence Relations, ⁋Definition 2) Then we may consider the canonical decomposition of \(f\)

\[X\overset{p}{\longrightarrow}X/R\overset{\bar{f}}{\longrightarrow}f(X)\overset{i}{\longrightarrow}Y\]

Now if we give \(f(X)\) the subspace topology, it is immediate from Proposition 4 and §Initial and Final Topology, ⁋Proposition 3 that \(\bar{f}\) is continuous. Also, by the definition of the canonical decomposition, \(\bar{f}\) is a bijection. In general \(\bar{f}\) need not be a homeomorphism (§Continuous Functions, ⁋Example 5), but the following holds.

Proposition 6 For the above diagram, the following are equivalent.

1. \(\bar{f}\) is a homeomorphism from \(X/R\) onto \(f(X)\).
2. For an open set \(U\subseteq X\) saturated with respect to \(R\), \(f(U)\) is open in \(f(X)\).
3. For a closed set \(C\subseteq X\) saturated with respect to \(R\), \(f(C)\) is closed in \(f(X)\).

This is obvious because the second or third condition exactly means that \(\bar{f}^{-1}\) is also continuous.

Meanwhile, in the same situation as above, suppose there exists a continuous section \(s:Y\rightarrow X\) of \(f\). Then \(f\) is surjective, so \(i=\id_Y\). Now both \(\bar{f}\) and \(p\circ s\) are continuous, and

\[\bar{f}\circ(p\circ s)=f\circ s=\id_Y\]

and composing \(\bar{f}^{-1}\) on the left and \(\bar{f}\) on the right of the above equation, respectively, we obtain

\[(p\circ s)\circ\bar{f}=\id_{X/R}\]

Therefore, in this case \(\bar{f}\) is a homeomorphism.

Quotient Spaces and Subspaces

Now consider a topological space \(X\), a subset \(A\), and an equivalence relation \(R\) given on \(X\). Letting \(p:X\rightarrow X/R\) be the canonical projection, the canonical decomposition of \(p\vert_A:A\rightarrow X/R\)

\[A\overset{q}{\longrightarrow}A/(R\vert_A)\overset{\overline{(p\vert_A)}}{\longrightarrow} f(A)\overset{j}{\longrightarrow}X/R\]

is defined, and by the same argument as above \(\overline{(p\vert_A)}\) is a continuous bijection. The following proposition is also almost obvious.

Proposition 7 In the above decomposition, the following are all equivalent.

1. \(\overline{(p\vert_A)}\) is a homeomorphism.
2. An open set \(U\subseteq A\) that is \(R\vert_A\)-saturated is the intersection of an open set in \(X\) that is \(R\)-saturated with \(A\).
3. A closed set \(C\subseteq A\) that is \(R\vert_A\)-saturated is the intersection of a closed set in \(X\) that is \(R\)-saturated with \(A\).

1. Just as in [§Subspaces], [Mun] takes this as the definition of the quotient topology. ↩