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I needed a way to point to the topological interval $[0,1]$ regarded as an interval object for the use in homotopy theory. Neither the entry interval nor the entry interval object seemed specific enough for this purpose, so I created topological interval.
I added Freyd’s characterization. (This is stated elsewhere, but also belongs here.)
Okay, thanks. Just for completeness, to make it the topological interval, I have added mentioning of the induced order topology to the statement of the theorem.
In some sense “interval” is a red herring because the functor could just as well be defined for bipointed topological spaces and one would get $[0, 1]$ with its standard topology. Maybe it’s a matter of taste how one puts it, but I used intervals because the article is “topological interval”.
Okay, but then we should not say
as an object of the category $Int$ of intervals (linear orders with distinct top and bottom elements)
but
as an object of the category $Int$ of intervals (linear orders with distinct top and bottom elements) equipped with their order topology
Not necessarily, because then we are speaking of the “category of topological intervals” or something. That’s more structural baggage than we actually need to carry around.
I probably could have made my point better by going minimal and removing both words, “topology” and “interval”. Just define $[0, 1]$ as the terminal $F$-coalgebra in the category of sets equipped with a distinct first and second point. Then the interval structure you want can be coinductively defined. More at coalgebra of the real interval.
All I want is that the entry titled “topological interval” says explicitly how this terminal coalgebra is a topological space, either way.
Okay, I reworded that section and hopefully it’s satisfactory now.
Okay, thanks!
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