  We prove some technical results on definable types in $p$-adically closed
fields, with consequences for definable groups and definable topological
spaces. First, the code of a definable $n$-type (in the field sort) can be
taken to be a real tuple (in the field sort) rather than an imaginary tuple (in
the geometric sorts). Second, any definable type in the real or imaginary sorts
is generated by a countable union of chains parameterized by the value group.
Third, if $X$ is an interpretable set, then the space of global definable types
on $X$ is strictly pro-interpretable, building off work of Cubides Kovacsics,
Hils, and Ye. Fourth, global definable types can be lifted (in a non-canonical
way) along interpretable surjections. Fifth, if $G$ is a definable group with
definable f-generics ($dfg$), and $G$ acts on a definable set $X$, then the
quotient space $X/G$ is definable, not just interpretable. This explains some
phenomena observed by Pillay and Yao. Lastly, we show that interpretable
topological spaces satisfy analogues of first-countability and curve selection.
Using this, we show that all reasonable notions of definable compactness agree
on interpretable topological spaces, and that definable compactness is
definable in families.
