Previously, I defined a path in terms of a continuous function from a closed interval on the reals to a set of points in a topological space.
Because the function is continuous, by definition, the resultant image is homeomorphic to the closed interval on the reals. Because any closed interval on the reals is itself homeomorphic to the specific closed interval [0, 1] then the image of a path can be said to be homeomorphic to the real interval [0, 1].
UPDATE (2005-06-01): As Michael Hudson points out in a comment, a path will only be homeomorphic to the closed interval [0, 1] if it doesn't cross over itself. Homeomorphisms require the function to be bijective, continuous and have a continuous inverse. A path that crosses over itself doesn't meet these criteria.
UPDATE: next post
by : Created on May 28, 2005 : Last modified June 1, 2005 : (permalink)
One challenge doing any kind of cross-disciplinary work is the differences in terminology. It's one thing where the same concept gets two different names—it's a lot harder when two different things get the same name.
I recently got into a confusing discussion on the b-greek mailing list where people (including some notable scholars) were saying things like "word order doesn't always alter syntax in Greek". As a linguist that sounds like utter contradiction but people insisted it was true for some constructions in Ancient Greek and "prominent linguists" had recognized this for decades.
I finally took my own advice and stepped back to look at the terminology being used. Then it struck me.
What gets called "syntax" by Greek scholars is largely what I would describe as mapping grammatical relations (e.g. SUBJECT) to semantic roles (e.g. AGENT). That's why when you open up a book on Greek "syntax" it spends a lot of time talking about what different cases means semantically.
This isn't what syntax means to a formal linguist or computer scientist. To them, "syntax" has to do with things like constituent structure and word order.
Now, in English, grammatical relations are predominantly determined by word order rather than morphology whereas in Greek, the word order matters less in determining grammatical relations and morphology takes on that role.
And here is the crux of the terminology confusion. Consider the previous paragraph. You could replace "word order" with "syntax" and it would mean (roughly) the same thing to a formal linguist. I suspect you could replace "grammatical relations" with "syntax" and it would mean the same thing to a Greek scholar.
So here's a way I suggested we could avoid confusion on the b-greek mailing list:
A. Whenever I see someone say "syntax", I'll read it as "grammatical relations".That way "word order doesn't always alter syntax in Greek" reads to me as "word order doesn't always alter grammatical relations" and I'll agree.
B. Whenever one sees me say "syntax", one should read it as "constituent structure, word order, etc"
That way "word order doesn't always alter syntax in Greek" reads as "word order doesn't always alter constituent structure, word order, etc in Greek" and you'll see why I think it's a contradiction.
by : Created on May 28, 2005 : Last modified May 28, 2005 : (permalink)