AM: morning, PM: afternoon/evening

Yeah, but this is the problem - why noon is "afternoon"?

"12:00:00 + ε" is after noon, but your clock will not show a difference.

Clopen intervals in order to have a well-specified time system.

[0] [12:00 PM, 12:00 AM) for after the noon instant

[1] [12:00 AM, 12:00 PM) for before the noon instant

24-hour time is superior to the mod-12 time system in almost every way but mechanical clock faces and soldier shifts under early systems that used sun dials. However, the equivalent question can be asked by a day is a clopen interval of [00:00 current day, 00:00 next day)

In 24-hour time, those are 12-24 and 0-12, respectively. https://en.wikipedia.org/wiki/ISO_8601

A place open 24 hours/day is open in the period 0-24.

I don't follow your point. Are you correcting my point on clopen intervals and mod bases?

I think clopen is too complicated in that it expresses ranges that are not practically useful, like (10,11], and it complicates expressing something like "Open 0-24", where your notation appears to require a date.

I think you get the essential utility you want in a 24 hour clock by simply letting 24 (or "24:00:00" when fully written out) be a way to represent the end of the day.

It's also already in use, with examples at https://ioi2023.hu/faq/ , http://www.garageferrarimilazzo.com/en/ , and https://jozsefvaros.hu/english/news-in-the-district/2022/12/... .

Noon isn't really before or after noon, but we want to lump it together with one of them, so we picked the one that made 12:00 and 12:01 go together. (Edit: or for that matter, 12:00:00 and 12:00:01, down to arbitrary closeness, as sibling comment points out)