Reply To: Colon in math equations
For the f: A → 𝒫(A), you might have already found the "function notation" question in this forum, which touches on the seven different mathematical uses for colons. I agree with your assessment that this is giving a name to a map, as defined by WolframMathWorld. As you said, in braille such a colon (for mapping) would be transcribed unspaced, and it would need a punctuation indicator.
For something like {x ∈ ℚ : x² < 2}, I agree with your assessment that the colon here means "such that." So, the expression {x ∈ ℚ : x² < 2} says, "x is an element of the set Q such that x-squared is less than 2." The "Colon and letters in sets of numbers" question in this forum talks about this "such that" role for a colon and recommends transcribing it unspaced (no blank braille cell before or after the colon meaning "such that").
Since we recommend transcribing both the mapping and "such that" colons completely unspaced, it may not be necessary for you to completely understand the difference between the two. For what it is worth, I agree with you that a "such that" colon is likely to be part of an expression within grouping symbols (most often curly braces). I would also suggest that a "such that" colon will have the same variable both before it and after it, like the x in our example expression above.
The print sign := (a colon followed by an equals sign) seems to mean "is defined as" (I first found this definition in a thread on StackExchange). How are you thinking about transcribing the := sign?
Regarding your concern about double-struck letters and the script P, what if you used the Nemeth Code script type form for the P's meaning "power set" and another Nemeth Code type form (italic, bold, or sanserif) for the letters that are double struck?
–Kyle