Vtyjkyuk

Let $Bbb P^1$ be complex projective space with the quotient topology from $Bbb C^2-0$, where we identify $(a,b)sim(c,d)$ if there exists $lambdainBbb C^times$ such that $lambda(a,b)=(c,d)$. Denote the equivalence class of $(a,b)$ by $[a:b]$.

Let $BbbCcupinfty$ be the one point compactification of $Bbb C$. I want to show that the map
$$gamma:BbbP^1toBbbCcupinfty$$
given by
$$gamma([1:b/a])=b/a$$
$$gamma([0:1])=infty,$$
is open.

I have shown that whenever $Usubset U_0subsetBbbP^1$ doesn't contain $[0:1]$ that it is open. So I am only interested in $U$ open in $BbbP^1$ containing $[0:1]$.

I denote by $U_0=[a:b]mid ane 0$.

So let $U$ be open in $BbbP^1$ such that $[0:1]in U$. Then $U=(Ucap U_0)cup [0:1]$, and thus $gamma(U)=gamma(Ucap U_0)cup gamma([0:1])=gamma(Ucap U_0)cupinfty$. Where previous calculations show $gamma(Ucap U_0)$ is open in $Bbb C$.

This means that (from the definition of the topology of the one point compactification) all that is left is to show that $gamma(Ucap U_0)^c$ is quasi-compact, but I can't seem to show this. Any help is greatly appreciated!

You need to show that $gamma(U)$ contains a neighborhood of $infty$. Such a neighborhood can be of the form $inftycup mathbbC-bar B(0,r)$, where $bar B(0,r)$ is the closed ball of radius $r$.

Since the topology of $mathbbP^1$ is the quotient topology, $U$ contains a ball $p(B((0,1),s)$ where $p:mathbbC^2-0rightarrow mathbbP^1$ is the quotient map. Let $zin mathbbC-bar B(0,s)$, $1over<s$ implies that $(1over z,1)in B((0,1);s)$, this implies that $mathbbC-bar B(0,s)subsetgamma(p(B(0,s))subsetgamma(U)$.