28 March 2008

Open and closed?

At Language Log there's a post about how English-speakers use "open" and "closed", which are not grammatically the same sort of thing, in opposition to each other -- "open"/"close" or "opened"/"closed" would, on the surface, make more sense. (Compare French ouvert and fermé, which are both past participles.)

I won't try to summarize the linguistic content of the post; I'm not a linguist, although I did go through a phase where that seemed interesting.

But in mathematics-land, open and closed aren't even opposites, in the sense that open means not-closed and closed means not-open. Of course the complement of an open set is closed, and vice versa, but that's a more complicated relationship, because now we're talking about two sets, not one. This is one of about a zillion examples of how we take perfectly good natural-language words and give them specific meanings (group, ring, field, set, class, ...), which may or may not be preferable to making up entirely new words as some other fields (biology comes to mind) prefer.


Anonymous said...

Question about open and closed sets (opened and closed sets? ha ha)

Is the empty set open? And its complement also open?


Michael Lugo said...


yes. That's the first of the axioms for a topology.

Anonymous said...

As a side-note, If you don't keep an open mind you are actually not being "closed-minded". You're being "close minded". Why? Because "close" is the opposite of "broad".

Anonymous said...

And of course only mathematicians would encourage such a butchering of the English language as "clopen".

Michael Lugo said...

I only found one google hit for ensemble fouvert, compared to 44,700 for ensemble ouvert. There are 6,360 for clopen set and 1.4 million for open set, a 220 to 1 ratio. Clearly French-speakers haven't made the analogous coinage in their language.

Anonymous said...

In re "clopen", it's a portmanteau. Along with "slithy", "mimsy", and "frumious", they pack two meanings into one word.

Those are just made-up words, you say? Well how about we discuss it over brunch? We could also discuss the architecture in Oxbridge. Whatever we come up with, we'll have to post on a blog.

To form a portmanteau is a perfectly cromulent construction.