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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

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

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

Why does it matter that hyperconnected is “used in general topology”? The usage in algebraic geometry is far, far more common. ~2025-37024-53 (talk) 08:22, 30 November 2025 (UTC)Reply

The notion being discussed here is a purely topological notion, which is used in general topology where is often goes by the name "Hyperconnected space". And it is also true that particular cases of that notion are used in algebraic geometry under the name of "irreducible space". But since the discussion itself is more general than algebraic geometry, it makes sense to leave it as is.

Note also that Irreducible space is a redirect to the same page. So nothing is lost if someone wants to link to this page using that name. PatrickR2 (talk) 05:03, 1 December 2025 (UTC)Reply

Naming conventions for wiki pages are not decided based on their mathematical generality, they’re decided based on frequency of use and recognizability. I asked 3 colleagues and not a single one knew the term “hyperconnected”. ~2025-37024-53 (talk) 05:50, 1 December 2025 (UTC)Reply

I would guess your colleagues were not topologists. PatrickR2 (talk) 08:12, 3 December 2025 (UTC)Reply

I have literally never heard anyone call such a space hyperconnected. Irreducible is the standard and should be used here. ~2025-37024-53 (talk) 08:21, 30 November 2025 (UTC)Reply

"Hyperconnected" is used quite a bit in general topology. PatrickR2 (talk) 04:57, 1 December 2025 (UTC)Reply

“in general topology” being a pretty much negligible portion of the mathematics taking place today ~2025-37024-53 (talk) 05:48, 1 December 2025 (UTC)Reply