Let \(H\) be a subgroup of the topological group \(G\). If \(x \in G\), define \(xH = \ts{x \cdot h \mid h \in H}\); this set is called a left coset of \(H\) in \(G\). Let \(G/H\) denote the collection of left cosets of \(H\) in \(G\); it is a partition of \(G\). Give \(G/H\) the quotient topology.
Show that if \(\alpha \in G\), the map \(f_\alpha(x) = \alpha \cdot x\) induces a homeomorphism of \(G/H\) carrying \(xH\) to \((\alpha \cdot x)H\). Conclude that \(G/H\) is a homogeneous space.
Show that if \(H\) is a closed set in the topology of \(G\), then one-point sets are closed in \(G/H\).
Show that the quotient map \(p: G \to G/H\) is open.
Show that if \(H\) is closed in the topology of \(G\) and is a normal subgroup of \(G\), then \(G/H\) is a topological group.
Solution
Let \(p\colon G\to G/H\) be the quotient map. A map \(g\colon G/H\to Z\) is continuous if and only if \(g\circ p\) is continuous, by the definition of the quotient topology. Left and right translations of \(G\) are homeomorphisms (E-1KV5G).
1. (a) The map \(\bar f_\alpha(xH)=(\alpha x)H\) is a well-defined homeomorphism of \(G/H\), and \(G/H\) is homogeneous.
\(f_\alpha(xH)=(\alpha x)H\), so \(\bar f_\alpha\) is well defined and \(\bar f_\alpha\circ p=p\circ f_\alpha\) is continuous; hence \(\bar f_\alpha\) is continuous. Its inverse is \(\bar f_{\alpha^{-1}}\), continuous for the same reason. Given cosets \(xH\) and \(yH\), the homeomorphism \(\bar f_{yx^{-1}}\) carries \(xH\) to \(yH\).
2. (b) If \(H\) is closed, one-point sets are closed in \(G/H\).
\(p^{-1}(\{xH\})=xH=f_x(H)\) is closed, as the image of the closed set \(H\) under a homeomorphism.
3. (c) \(p\) is open.
For open \(U\subseteq G\), \(p^{-1}(p(U))=UH=\bigcup_{h\in H}Uh\) is a union of right translates of \(U\), hence open, so \(p(U)\) is open.
4. (d) If \(H\) is closed and normal, \(G/H\) is a topological group.
Since \(H\) is normal, \(G/H\) is a group with multiplication \(\bar m(xH,yH)=xyH\) and inversion \(\bar\iota(xH)=x^{-1}H\), and \(\bar m\circ(p\times p)=p\circ m\), \(\bar\iota\circ p=p\circ\iota\). By step 3, \(p\times p\) is a continuous open surjection, hence a quotient map, so \(\bar m\) is continuous; \(\bar\iota\) is continuous because \(\bar\iota\circ p\) is. By step 2, \(G/H\) is \(T_1\).
5. Q.E.D.
Steps 1 through 4 prove (a) through (d).