I noticed that in some posts, the code of the equations ends with "\:.", but nor "\:", nor ":" themselves appear in the rendered equation (see, e.g., this answer). However, if I copy-paste such an equation, e.g.
$$V(t)U_t = U_t V(0) \quad \forall t \in \mathbb R\:,\tag{1}$$
into the editor of a new question, ":" is shown literally also in the rendered equation seen below the editor window. I see that "\:" represents a kind of space character, but why does it behave differently when I edit a post or add a question?
(I noticed also that in this question, the MathJax code isn't processed at all. Why?)
\:produces a space. For instance, see egreg's answer to "Math mode horizontal space difference between \ (backslash) and \; (backslash semicolon) and what to use for multiplication" on TeX - LaTeX SE. If you want the colon to be rendered then you shouldn't use the backslash.\:? The tutorial seemingly only lists\,(backtick + comma. Thin space) and\;(backtick + semicolon. Wider space). And\quadand\qquad. And implicitly escaping space (\)? Perhaps they mistyped\;as\:?\:is equivalent to the spacing that is inserted around a binary operator, as per the link I shared in a comment above.