For which properties $(P)$ is the following statement known to be true?
In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the limit of a convergent sequence of $(P)$-elements has $(P)$.
For example, 'unimodular' (conjugating by the element preserves the Haar measure) is such a property.
I'm also interested in the analogous question just for the class of totally disconnected locally compact groups (e.g. 'uniscalar' is such a property for t.d.l.c. groups, but I don't think it has been defined for general locally compact groups).