The real projective line and the action of SL2(R)

Give ${\mathbb{R}}^2$ its finite product topology and define $P^1(\mathbb{R})$ to be the set of equivalence classes of nonzero pairs $(x,y)\in {\mathbb{R}}^2$, where $(x,y)\sim (\lambda x,\lambda y)$ for $\lambda \in {\mathbb{R}}^{\times }$; write a class as $[x:y]$. Identify it as a set with the one-point compactification ${\mathbb{R}}^{\ast }=\mathbb{R}\cup \{\mathrm{\infty }\}$ (The one-point (Alexandroff) compactification $X^{\ast }=X\cup \{\mathrm{\infty }\}$, whose open sets are the open sets of $X$ together with the complements in $X^{\ast }$ of the closed compact subsets of $X$) by $[t:1]\leftrightarrow t$ and $[1:0]\leftrightarrow \mathrm{\infty }$, and give it the transported topology. The map $h(t)=\frac{2t}{1+t^2}+i\frac{t^2-1}{1+t^2}(t\in \mathbb{R}),h(\mathrm{\infty })=i$ is a homeomorphism ${\mathbb{R}}^{\ast }\to \mathbb{T}:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$; consequently $P^1(\mathbb{R})$ is compact and metrizable (The multiplicative unit circle is a compact metrizable topological abelian group, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not). The map $\phi :{\mathbb{R}}^2\setminus \{(0,0)\}\to P^1(\mathbb{R})$, $\phi (x,y)=[x:y]$, is continuous and surjective, and $\phi (u,v)=\phi (x,y)$ exactly when $(u,v)=\lambda (x,y)$ for some $\lambda \in {\mathbb{R}}^{\times }$ (Continuity of a map of topological spaces at a point and globally).

Let $G={\mathrm{S}\mathrm{L}}_2(\mathbb{R})=\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):ad-bc=1\}$ have matrix multiplication and the subspace topology from the finite product ${\mathbb{R}}^4$ (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, Topological group: multiplication and inversion are continuous). For $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in G$, define $g\cdot [x:y]:=[ax+by:cx+dy]\mathrm{.}$ This is a well-defined left action by homeomorphisms, and its action map $G\times P^1(\mathbb{R})\to P^1(\mathbb{R})$ is continuous (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological). In the coordinate $t=x\mathrm{/}y$, $g\cdot t=\frac{at+b}{ct+d}(ct+d\mathrm{\ne }0),g\cdot (-d\mathrm{/}c)=\mathrm{\infty }(c\mathrm{\ne }0),g\cdot \mathrm{\infty }=\{\begin{array}{ll}a\mathrm{/}c,& c\mathrm{\ne }0,\\ \mathrm{\infty },& c=0.\end{array}$ In particular $u_{+}=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ acts by $t\mapsto t+1$ and fixes $\mathrm{\infty }$. The matrix $u_{-}=\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)$ acts by $t\mapsto t\mathrm{/}(t+1)$, fixes $t=0$, and sends $t=-1$ to $\mathrm{\infty }$. In the other chart $s=y\mathrm{/}x$, $u_{-}$ acts by $s\mapsto s+1$ for finite $s$ and fixes the omitted point $s=\mathrm{\infty }$ (the line $[0:1]$).

For the circle map, $(2t{)}^2+(t^2-1{)}^2=(1+t^2{)}^2$, so $\mathrm{\mid }h(t)\mathrm{\mid }=1$. If $z=x+iy\mathrm{\ne }i$ lies on the circle, then $1-y\mathrm{\ne }0$ and $t=x\mathrm{/}(1-y)$ satisfies $h(t)=z$; the remaining circle point $i$ is $h(\mathrm{\infty })$. At finite $t$ the coordinates of $h$ are quotients of continuous real functions with denominator $1+t^2>0$. Also $\mathrm{\mid }h(t)-i\mathrm{\mid }=2\mathrm{/}\sqrt{1+t^2}\to 0$ as $\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty }$. For each $\eta >0$, choose $R>0$ with $2\mathrm{/}\sqrt{1+R^2}<\eta$. The set ${\mathbb{R}}^{\ast }\setminus [-R,R]$ is a neighbourhood of $\mathrm{\infty }$, since $[-R,R]$ is compact by Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact and closed by A compact subset of $\mathbb{R}$ is closed and bounded, and $h$ maps it into the $\eta$-ball about $i$. Thus $h$ is continuous at $\mathrm{\infty }$. It is therefore a continuous bijection from compact ${\mathbb{R}}^{\ast }$ ($X^{\ast }$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{\ast }$ exactly when $X$ is not compact; and $X^{\ast }$ is Hausdorff exactly when $X$ is locally compact and Hausdorff) to the Hausdorff metric circle (Distinct points of a metric space have disjoint balls around them), so it is a homeomorphism by A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Pulling back the circle metric gives a metric on $P^1(\mathbb{R})$.

The map $\phi$ is continuous at every point with $y\mathrm{\ne }0$: near $(x,y)$ keep $\mathrm{\mid }{y}^{\mathrm{\prime }}\mathrm{\mid }>\mathrm{\mid }y\mathrm{\mid }\mathrm{/}2$, and use $\mid x^{\mathrm{\prime }}\mathrm{/}{y}^{\mathrm{\prime }}-x\mathrm{/}y\mid \le 2\mathrm{\mid }{x}^{\mathrm{\prime }}-x\mathrm{\mid }\mathrm{/}\mathrm{\mid }y\mathrm{\mid }+2\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }{y}^{\mathrm{\prime }}-y\mathrm{\mid }\mathrm{/}\mathrm{\mid }y{\mathrm{\mid }}^2$. If $y=0$ then $x\mathrm{\ne }0$; for any closed compact $K\subseteq \mathbb{R}$, choose $M$ with $\mathrm{\mid }t\mathrm{\mid }\le M$ for all $t\in K$ (A compact subset of $\mathbb{R}$ is closed and bounded). On the neighbourhood $\mathrm{\mid }{x}^{\mathrm{\prime }}-x\mathrm{\mid }<\mathrm{\mid }x\mathrm{\mid }\mathrm{/}2$, $\mathrm{\mid }{y}^{\mathrm{\prime }}\mathrm{\mid }<\mathrm{\mid }x\mathrm{\mid }\mathrm{/}(2(M+1))$, a point with $y^{\mathrm{\prime }}=0$ maps to $\mathrm{\infty }$, while a point with $y^{\mathrm{\prime }}\mathrm{\ne }0$ has $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{/}{y}^{\mathrm{\prime }}\mathrm{\mid }>M$ and maps outside $K$. This proves continuity at $[x:0]$ by the neighbourhood basis of the one-point compactification. The fiber statement follows by comparing ratios when both second coordinates are nonzero; when the common value is $\mathrm{\infty }$, both second coordinates vanish and the first coordinates are nonzero, so the pairs are again nonzero scalar multiples.

The set $G$ is a group: determinant multiplicativity gives closure under multiplication, and the inverse of $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is $\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$; associativity is inherited from matrix multiplication. Its multiplication entries are sums of products of coordinate maps, and inversion entries are coordinate maps with signs, so both are continuous by Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined and the finite product and subspace topologies. Thus $G$ is a topological group without using a choice-dependent Lie-group result.

The reciprocal map $J:{\mathbb{R}}^{\ast }\to {\mathbb{R}}^{\ast }$, $J(t)=1\mathrm{/}t$ for $t\mathrm{\ne }0,\mathrm{\infty }$, $J(0)=\mathrm{\infty }$, and $J(\mathrm{\infty })=0$, is a homeomorphism. It is continuous away from $0,\mathrm{\infty }$ by ordinary reciprocal continuity. Given a neighbourhood ${\mathbb{R}}^{\ast }\setminus K$ of $\mathrm{\infty }$, choose $M$ bounding the compact set $K$ (A compact subset of $\mathbb{R}$ is closed and bounded); then $J$ maps $(-1\mathrm{/}(M+1),1\mathrm{/}(M+1))$ into that neighbourhood, proving continuity at $0$. Every interval $(-\epsilon ,\epsilon )$ about $0$ contains $J(t)$ for $\mathrm{\mid }t\mathrm{\mid }>1\mathrm{/}\epsilon$ and for $t=\mathrm{\infty }$, and that tail is a neighbourhood of $\mathrm{\infty }$, proving continuity there. Since $J^2$ is the identity, $J$ is a homeomorphism. It is the coordinate swap $[x:y]\mapsto [y:x]$ and supplies the second coordinate chart $s=y\mathrm{/}x$ around $t=\mathrm{\infty }$.

For joint continuity of the action, use the two source charts $[t:1]$ and $[1:s]$. Their unnormalized output coordinates are $(at+b,ct+d)$ and $(a+bs,c+ds)$, respectively. They cannot both vanish because $g$ is invertible. Wherever the second coordinate is nonzero, the target $t$-coordinate is the quotient of the first by the second; wherever the first is nonzero, the target $s$-coordinate is the quotient of the second by the first. These quotients are continuous on their open domains by Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined. The charts cover the source and target, so the action map is continuous. The identity and composition laws follow from matrix multiplication, and the map for $g^{-1}$ is the inverse homeomorphism. No choice principle is used.