Timeline for Is the empty graph a tree?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 19, 2022 at 17:36 | comment | added | PrimeRibeyeDeal | How about the term "nondisconnected" to mean connected or empty? | |
| Apr 17, 2020 at 18:02 | comment | added | Todd Trimble | Normally when people define partitions of a set, they mean a collection of disjoint nonempty subsets $X_1, X_2, \ldots$ that cover the set, as opposed to a situation where some of the partition classes are allowed to be empty (they'd still be disjoint, after all). And people want there to be a bijective correspondence between equivalence relations and partitions. That's another way to think of it. | |
| Apr 17, 2020 at 17:55 | comment | added | Todd Trimble | If you want the quotient map from a set $X$ to the set of equivalence classes $X/E$ with respect to an equivalence relation $E$ to be a surjection (and I think theorems become more awkward if you don't), then yes, equivalence classes had better be nonempty. For example, if you want quotients and disjoint sums to get along properly, then it'd be a little weird to consider a disjoint sum of sets equipped with equivalence relations (inducing an equivalence relation on the whole), and then consider the case where a lot of those sets are empty. I feel confident my convention is "best practice". | |
| Apr 17, 2020 at 17:30 | comment | added | user56097 | I usually see connectedness to be defined before the notion of connected components. And usually, for this original definition, empty spaces are indeed connected. But maybe this says more about the kind of maths books I read than about mathematics itself... :-/ | |
| Apr 17, 2020 at 17:27 | comment | added | user56097 | @ToddTrimble Is it 100% uncontroversial that an equivalence class has to be nonempty? If one defines an equivalence class as a maximal subset where any two points are in relation, it can be empty if the ambient set is empty. | |
| Jan 28, 2014 at 3:46 | history | edited | André Henriques | CC BY-SA 3.0 |
added 238 characters in body
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| Feb 5, 2013 at 0:07 | comment | added | Todd Trimble | Quite so. Moreover, a connected component is an equivalence class, and an equivalence class of an equivalence relation on a set is by definition nonempty. Therefore... | |
| Feb 4, 2013 at 23:26 | history | answered | André Henriques | CC BY-SA 3.0 |