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Comment by orlp | Hacker News Reader

𝕏 is used as a symbol for a generic metric space. E.g. if your result is generic over ℝ², ℂ³, etc, you can write 𝕏.

I'm a PhD student in algebraic geometry and have seldom seen this. IME "X" is a generic name for all kinds of spaces (metric, topological, complex analytic, algebraic...), but I don't remember ever having seen "𝕏".

That said, in principal everyone is free to call their spaces how they like. It just makes it easier to read if one sticks to familiar notation.

You're right I should've used "as a symbol" not "as the symbol". It's not terribly common.

Now that I think about it, most double stroked characters really represent "concrete" objects, not generic ones. Like ℙ for projective space, or 𝔸 for affine space. Also 𝔽 is often used together with an index 𝔽_q for the field with q elements.

On the other hand, Wikipedia[1] agrees with you, that 𝕏 is "occasionally" used for metric spaces.

[1] https://en.wikipedia.org/wiki/Blackboard_bold

In set theory blackboard bold P, Q and C are very often used for arbitrary (forcing) posets

As a counterpoint, 𝔽 when not fixed to a particular size is arguably still generic over "any field", and 𝔾 is most definitely generic for "any group", even if fixed to a particular size.

There might be small cultural differences at hand here. As a German, I prefer "k" or "K" for fields (as in "Körper"), and I think I've seen 𝔾 mostly as 𝔾_a or 𝔾_m for the additive and multiplicative group. shrug

My experience with 𝔽 is “vector space over a field 𝔽”, which is not a concrete object, but I have probably seen F written in a normal italic style more often.

I've seen (and used) \mathbb{P} and \mathbb{E} for the probability & expectation operators (so not really a "concrete" object in your sense?)

I've seen ℙ used for probability.

I’ve also seen ⅆ as the differential operator and 𝔼(X) as the expected value of X. I think I’ve also seen 𝟙_x, and ⅈ, and ⅉ but I don’t remember for what.

I know 𝟙_X as the indicator function (also called characteristic function) of X in measure theory and probability theory, but I’ve also seen 1_X or χ_X.

ⅈ and ⅉ can be basis vectors or imaginary units although I’ve never seen them in blackboard bold in my lectures.

Thanks! I am always a little curious about how these symbols are used. That makes perfect sense to me.

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