  We consider Galton-Watson branching processes with countable typeset
$\mathcal{X}$. We study the vectors ${\bf q}(A)=(q_x(A))_{x\in\mathcal{X}}$
recording the conditional probabilities of extinction in subsets of types
$A\subseteq \mathcal{X}$, given that the type of the initial individual is $x$.
We first investigate the location of the vectors ${\bf q}(A)$ in the set of
fixed points of the progeny generating vector and prove that $q_x(\{x\})$ is
larger than or equal to the $x$th entry of any fixed point, whenever it is
different from 1. Next, we present equivalent conditions for $q_x(A)< q_x (B)$
for any initial type $x$ and $A,B\subseteq \mathcal{X}$. Finally, we develop a
general framework to characterise all \emph{distinct} extinction probability
vectors, and thereby to determine whether there are finitely many, countably
many, or uncountably many distinct vectors. We illustrate our results with
examples, and conclude with open questions.
