Composition and continuity of game coverings

For identity maps, take a target play itself as its lift; every condition in F1 is then an equality. For the composite, prefix and length preservation compose. If ${\pi }_1{\pi }_2(r)$ is taboo for a player, first reflection makes ${\pi }_2(r)$ taboo for that player and second reflection makes $r$ so. Same-player strategy preservation also composes. If two input strategies agree below depth $n$, locality for ${\varphi }_2$ makes their images agree below $n$, and locality for ${\varphi }_1$ does so once more.

Given maximal $x$ consistent with ${\varphi }_1{\varphi }_2(\sigma )$, first lift it to maximal $y$ on $S$ consistent with ${\varphi }_2(\sigma )$, then lift $y$ to maximal $z$ on $R$ consistent with $\sigma$. We have ${\pi }_1(y)\subseteq x$ and ${\pi }_2(z)\subseteq y$, whence ${\pi }_1{\pi }_2(z)\subseteq x$. If both lifts project exactly, so does the composite. If the second is proper, $z$ is taboo for $P$. If only the first is proper, $y$ is taboo for $P$ and ${\pi }_2(z)=y$; taboo reflection then makes $z$ taboo for $P$. These exhaust the alternatives and prove composite lifting.

Set $k=\min (k_1,k_2)$. The three trees' nodes and labels agree through depth $k$, and both position maps are identity there; their composite is identity there. Both strategy maps preserve every prescribed move below $k$, so their composite does too. This proves the $k$-covering clause, including $k=0$.

For any covering and target node $p$, length and prefix preservation give ${\pi }^{-1}([T{]}_p)=\bigcup \{[S{]}_s:\mathrm{\mid }s\mathrm{\mid }=\mathrm{\mid }p\mathrm{\mid },\pi (s)=p\}$. The right side is open, hence preimages of all unions of cylinders are open. If $B$ and its complement are open, their preimages are open and complementary in $[S]$. Thus the preimage of $B$ is clopen, completing the assertions. QED.