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• CommentRowNumber1.
• CommentAuthorAndrew Stacey
• CommentTimeJun 22nd 2010

Just got the following query from Harald Hanche-Olsen about the page separation axioms. As I’ve never seen that notation before either (but agree with Harald’s comments in both parts), I’m forwarding it here so that the person who first adopted it (Toby?) or others can chip in.

I hadn’t seen the notation $\stackrel\circ\ni$ for a neighbourhood before, but it looks like a reasonable notation that I might want to adapt. BUT it seems more appropriate for a neighbourhood of a point rather than a neighbourhood of a set. Wouldn’t $\stackrel\circ\supset$ or $\stackrel\circ\supseteq$ be more appropriate for that case? What is the rationale for the usage on that page?

• CommentRowNumber2.
• CommentAuthorHarry Gindi
• CommentTimeJun 22nd 2010

Unicode has glyphs for open subset and open superset

• CommentRowNumber3.
• CommentAuthorAndrew Stacey
• CommentTimeJun 22nd 2010

What are they?

Not sure they’d be right here because we don’t want to say “$U$ is an open subset of $X$” but rather “$U$ is a neighbourhood of $F$”. I’m guessing that the notation is meant to suggest “$F$ is a subset of the interior of $U$”.

• CommentRowNumber4.
• CommentAuthorHarry Gindi
• CommentTimeJun 22nd 2010
• (edited Jun 22nd 2010)

I can’t actually write them down (and I tried to make them in LaTeX, but I gave up). It’s a \circ in the middle of a \subset (don’t bother trying to make it in LaTeX, the symbols are aligned differently and can’t be composed without doing some serious work with hboxes and vboxes).

• CommentRowNumber5.
• CommentAuthorAndrew Stacey
• CommentTimeJun 22nd 2010

Ah, I see. You mean ⟃ (and ⟄). So one would write $U ⟃ X$ for “$U \subset X$ and $U$ open”.

(To coin a phrase: iTeX rocks, dude)

• CommentRowNumber6.
• CommentAuthorHarry Gindi
• CommentTimeJun 22nd 2010
• (edited Jun 22nd 2010)

Yeah. There’s also a symbol for “closed subset”.

• CommentRowNumber7.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2010

Yes, ‘$\stackrel\circ\ni$’ should be for a neighbourhood of a point, not for a neighbourhood of a set. (And as far as I know, it’s due to me.) For a neighbourhood of a set, ‘$\stackrel\circ\supseteq$’ would be good (and is due to Harald).

• CommentRowNumber8.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2010

I’ve fixed the article.

• CommentRowNumber9.
• CommentAuthorHarry Gindi
• CommentTimeJun 23rd 2010

Hey, Toby, you realize that $\ni$ means “such that”, right?

• CommentRowNumber10.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2010

@ Harry:

It can, but it also means ‘owns’; that is, $A \ni x$ if and only if $x \in A$. Furthermore, there is a fuller form of the ‘such that’ symbol that allows you distinguish it from the ‘owns’ symbol. (Sorry I cannot find it in Unicode, but it amounts to continuing the horizontal line farther to the right.) Finally, there is no need for a symbol that stands for ‘such that’ in the first place! For example, there is no reason to write ‘$\exists x \ni P(x)$’, you just write ‘$\exists x\, P(x)$’. (Sometimes it’s nice to have some punctuation here, such as a comma, but that holds just as well with ‘$\forall$’.)

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeJun 23rd 2010

I detest the usage of $\ni$ to mean “such that.” (Although I mean that in the nicest possible way, of course. (-: ) I think mathematics makes much more sense if writing a relation symbol backwards means reversing the variables in the relation, cf. $x\lt y$ and $y\gt x$.

I usually pronounce $\ni$ as “contains,” although I guess that could also be used to mean $\supseteq$. Fortunately, in ETCS an element of a set is exactly the same as a singleton subset. (-:

• CommentRowNumber12.
• CommentAuthorTodd_Trimble
• CommentTimeJun 23rd 2010

Maybe I’m just making this up, but I thought I remembered Freyd calling it “ni”, as in the knights who say… ni!

• CommentRowNumber13.
• CommentAuthorGuest
• CommentTimeJun 23rd 2010
Well, ∋ is written \ni in TeX …

- Harald H-O
• CommentRowNumber14.
• CommentAuthorHarry Gindi
• CommentTimeJun 23rd 2010
• (edited Jun 23rd 2010)

I detest the usage of $\ni$ to mean “such that.” (Although I mean that in the nicest possible way, of course. (-: ) I think mathematics makes much more sense if writing a relation symbol backwards means reversing the variables in the relation, cf. $x\lt y$ and $y\gt x$.

Blame Stephen DeBacker for getting my freshman class hooked on $\ni$.

• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeJun 23rd 2010

I just remembered that and came back to admit my failure of memory, but you beat me to it, Harald! Actually, Freyd pronounces it “epsiloff”. :-)

• CommentRowNumber16.
• CommentAuthorAndrew Stacey
• CommentTimeJun 24th 2010

Back on track, I really like these notations for (essentially) “is in the interior of” or “the interior contains” for both an element or a subset. The link to the usual notation of the interior of a set makes it really easy to remember.

Plus I can tell my topology students that this notation has been (partially) forged by someone in my own department! Who says mathematicians don’t do anything useful?

Now all we need to do is get them into the unicode list. There’s ⫃ (and ⫄, ⋵, but no reverse of that one.) For the moment, I guess that the stackrel command produces a decent combination.

Incidentally, Toby, did you mean ⋲ (and ⋺)? While I’m searching, here’s the closed subset notation (according to unicode): ⫏ ⫐ ⫑ ⫒

There’s also standard entity ways of putting accents on top of letters; so a similar effect could be obtained by putting the appropriate accent on top of the subset symbol (this would be quite appropriate given Harald’s nationality!). So we could have ⊂ͦ and ⊃ͦ and ∊ͦ. Hmm, not quite centred on the “contained in” symbol there. Not sure how this would work in maths, though: $F \mathop{⊂ͦ} U$ versus $F \stackrel\circ\subset U$. No, the unicode version either gets separated or gets mangled a bit whatever I try.

• CommentRowNumber17.
• CommentAuthorTobyBartels
• CommentTimeJun 24th 2010
• (edited Jun 24th 2010)

The accents aren’t really expected to work, since you’re putting an alphabetical accent on top of a mathematical symbol. They’ll look nice only if some coincidence in the fonts pertains. The Computer Modern fonts were, in part, designed to allow this sort of thing to work, but Unicode fonts generally aren’t.

So <mover> (that’s math over) in MathML (which is what iTeX’s \stackrel and \overset produce) is probably the correct way to do it.

Incidentally, Toby, did you mean ⋲ (and ⋺)?

Thank you, Andrew, I meant ‘⋺’, but I couldn’t find it.

By the way, am I the only one using a font on the Forum in which ‘⟃’ and its partner don’t look right? (‘⫑’ and its variants look fine.) Here is how it should look:

I like these symbols too, although they serve a different purpose.

• CommentRowNumber18.
• CommentAuthorAndrew Stacey
• CommentTimeJun 24th 2010

By the way, am I the only one using a font on the Forum in which ‘⟃’ and its partner don’t look right?

What font are you using? I’ve installed a firefox plugin that tells me exactly which font a particular glyph is being rendered in. For ⟃ (in text mode) it tells me that I’m using Trebuchet MS. It looks close enough to the picture that you drew - perhaps a little stretched horizontally, but not different enough to make me think that’s what you’re on about. It might be different in maths mode: $⟃$. Yes, there I get STIXGeneral. The Trebuchet one is a little lighter and so the circle seems further away from the enclosing symbol.

• CommentRowNumber19.
• CommentAuthorTobyBartels
• CommentTimeJun 24th 2010

I don’t think that I’m using any different fonts from whatever the Forum says to use. (Unless the Forum doesn’t say, in which case I seem to be using a default X sans-serif font, or DejaVu Serif in MathML.) On the Lab, I use DejaVu Sans outside MathML; that also doesn’t look right. The same bad look appears in MathML (DejaVu Serif) and fixed pitch (some default X monospace).

Incidentally, the bad look in question is that the circle is the same height as the subset symbol, but centred farther to the left, so that it looks somewhat like what $O\!\!C$ would produce in TeX.

What plugin is this that identifies fonts? (And is there a plugin that identifies Unicode code point? I would kill for that!)

• CommentRowNumber20.
• CommentAuthorAndrew Stacey
• CommentTimeJun 24th 2010

One such is FontFinder:

Searching for “font” in the firefox addons and then filtering to “Web Developer” gives a few others, one of which shows the font in the right-click context menu, and another which displays it in firebug.

The forum does suggest fonts, but if you don’t have the top one then it might be displaying a different one.

• CommentRowNumber21.
• CommentAuthorTobyBartels
• CommentTimeJun 24th 2010

Thanks, Andrew!

FontFinder says that the font on the problematic symbol is Trebuchet MS, or STIXGeneral in math mode.

• CommentRowNumber22.
• CommentAuthorAndrew Stacey
• CommentTimeJun 24th 2010

Strange, that’s what I was told too. Did it look wrong in both text and maths mode? Have you installed the recent release of the stix fonts?

• CommentRowNumber23.
• CommentAuthorTobyBartels
• CommentTimeJun 24th 2010

Did it look wrong in both text and maths mode?

Yes.

Have you installed the recent release of the stix fonts?

No, unless my package manager suggested it to me automatically, in which case I did. But that wouldn’t explain the text-mode mismatch.