Empty set?

Is the empty set considered to be irreducible? Hartshorne excludes it from the definition in his book Algebraic Geometry (page 3 in my edition). Nielius (talk) 13:22, 9 June 2012 (UTC)[reply]

I don't know, but I think that this is a relevant question, if only because I've opened this talk page precisely meaning to ask it myself! - Saibod (talk) 08:33, 26 December 2014 (UTC)[reply]

The empty space is certainly *not* irreducible. (And it is also *not* connected.) This should be corrected. (Not just on Wikipedia, by the way.) One good reference is e.g. Grothendieck, Dieudonné, EGA I (1971), Chapter 0, § 2. ---oo- (talk) 21:24, 24 November 2016 (UTC)[reply]

The truth is that it depends on the author. As a rule of thumb, more modern authors and authors that are closer to the use of the concept in algebraic geometry don't consider the empty set irreducible. Older text, texts that are closer to topology, or simply that didn't consider the question, consider it irreducible. In this cases it is not that the empty set was explicitly called irreducible, but that the definition given just applied to it. This article, and also Irreducible_component should probably mention that state of affairs, that there is not a unique definition in use. (My speculation is that with time people will settle with the definition that the empty set is not irreducible). Thatwhichislearnt (talk) 19:33, 23 January 2024 (UTC)[reply]

Like you say, in general topology the simplest thing to do is to not make a special case for the empty set. And then the empty set happens to be hyperconnected/irreducible according to the definition. But there is nothing particularly interesting about the empty set, so who cares (in general topology at least) whether it's irreducible or not. It just happens to be hyperconnected according to the simpler definition. PatrickR2 (talk) 07:02, 25 January 2024 (UTC)[reply]

But in algebraic geometry, the Zariski topology in particular, there may be a specific reason why it is more convenient to assume the empty set is not irreducible/hyperconnected. Maybe some theorems in algebraic geometry are simpler to state with that assumption, but it's not my area of expertise. Do you know what that reason could be? PatrickR2 (talk) 07:05, 25 January 2024 (UTC)[reply]

Considering 1 a prime or not, empty set connected or not, and empty set irreducible or not, have analogous advantages and disadvantages. Here https://ncatlab.org/nlab/show/empty+space and https://ncatlab.org/nlab/show/too+simple+to+be+simple they talk a bit about it. For example, in factorization/decomposition. Do we allow any number of factors 1 in the prime factorization? We could. The uniqueness in unique factorization already has the caveat about ordering of the factors. It is more convenient to have fewer clarifications, though. Same happens with decomposition into connected and irreducible components. That's one example. While the choice for 1 is pretty much settled, both choices are in use for the empty set. That's fine. Wikipedia just needs to clarify everywhere. Thatwhichislearnt (talk) 14:30, 25 January 2024 (UTC)[reply]

Definition

I think the closed sets in the definition should not be allowed to be equal to X itself.--Lieven Smits (talk) 11:33, 4 December 2015 (UTC)[reply]

@Lieven Smits: This edit replaced "non-empty" with "proper". GeoffreyT2000 (talk) 14:59, 4 December 2015 (UTC)[reply]

Thanks Geoffrey!--Lieven Smits (talk) 08:23, 11 March 2016 (UTC)[reply]

'Hyperconnected' or 'irreducible' in the article title?

What towns of mathematicians use the terminology 'hyperconnected' instead of 'irreducible'? I come from the algebraic geometry town, and there people only use the word 'irreducible.' After several years doing math, the only place where I've ever heard or read the term 'hyperconnected' is this wikipedia article. What about changing the article title to 'irreducible space'? Eliasgv3 (talk) 13:41, 8 March 2023 (UTC)[reply]

The notions are indeed the same, but hyperconnected" is used in general topology. So let us not change the title. Also constrast it with ultraconnected space. For more algebraic geometry related, see also Irreducible component. PatrickR2 (talk) 22:36, 8 March 2023 (UTC)[reply]