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Sometimes I use hair spaces to quickly try out a new kerning. It's also fun to change your tracking from time to time.
This is like a success story, replacing a legacy typesetting system with `most` of its features into a digital equivalent. So much harder to achieve with business processes in general.
As a web dev with a secret love for typography, I use these all the time in similar situations e.g. the hair space around em-dashes. I'm also trying, where possible, to ensure our CMS auto-corrects just for things like em-dashes in the first place; our writers find it difficult to use all the typographically correct punctuation, particularly when keyboard support is fairly weak.

On that issue (sort of), which text editor is best for supporting this kind of thing? Ideally, I want something that a) makes it easy to insert Unicode characters e.g. based on a name lookup b) more clearly identifies the exact Unicode character my cursor is on. It would be useful if, in both cases, common punctuation (these spaces, dashes, quotes) were flagged particularly 'well', but a general solution would be a good start.

One approach that I've used in intellij when writing HTML templates is to use the live templates to replace a string with the right html entity.

So I can do thinsp<tab> and it'll replace with &#8202;

Lots of editors support something like live templates so that would be one solution (if you are writing the HTML directly).

Emacs has the interactive built-in function insert-char (C-x 8 RET). You are prompted for the hex value or the Unicode name. It has autocompletion via the TAB key, so you can easily navigate through the Unicode names. Use describe-char to get info about the character you are currently on. See also Xah's page: http://ergoemacs.org/emacs/emacs_n_unicode.html
It's a sad implication of the publishing model in physics that I'm very familiar with most of the ideas in this piece. Academia cries out for a successor to LaTeX.
Interesting. Could these different space character alter the meaning of a sentence in other languages and not just the presentation? Example:

  "carpet" !== "car pet" but "car       pet" === "car pet".
The space between two letters can be described by 1 bit. Are there non-latin alphabets where it would take more than 1 bit to represent this information?
I enjoyed this little foray into the wilds of the Unicode(TM) landscape and typographic emptiness. As an aside, my interest in typography is matched by my interest in maps, and I've met numerous other men who share an interest in both typography and maps - is it a male trait, and what's the link between the two (if any)?
I learned to like em and en dashes as grammatical devices but was undecided about the space around them — "regular" spaces are too much, but having them flush with the surrounding text was equally undesirable. The "hair" space fits the bit fantastically, and it doesn't require HTML, CSS, or any special fonts!