Timeline for Interesting examples of vacuous / void entities
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 21, 2020 at 9:13 | comment | added | Georges Elencwajg | Glad you noticed, dear @LSpice! | |
| Dec 20, 2020 at 21:42 | comment | added | LSpice | "Quite categorical", heh. | |
| Jul 9, 2020 at 19:39 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 15 characters in body
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| Apr 14, 2015 at 18:08 | comment | added | Martin Brandenburg | Oups, I meant continuous functions into discrete spaces, aka locally constant functions. | |
| Oct 19, 2014 at 18:06 | comment | added | Martin Brandenburg | @AndreasBlass: Wow, this is a very concise definition! We could also say that on a connected space every continuous function should be constant of some value (i.e. factors over $\{\star\}$). But functions on $\emptyset$ have no value. | |
| Nov 15, 2010 at 12:48 | comment | added | Georges Elencwajg | Dear ACL: yes, I also like the idea of restricting the notion of (universal) covering space to pointed spaces. I vaguely remember reading that our Overlords (Deligne et al.) are quite categorical on this point. | |
| Nov 15, 2010 at 4:24 | comment | added | David Roberts♦ | There is a weakly universal covering, though: it can be dscribed as the colimit over fundamental groupoid with values the usual pointed universal covers. This all assumes these covers exist, which requires conditions on the base etc etc.. | |
| Nov 14, 2010 at 23:48 | comment | added | ACL | @Georges: In the category of spaces, there can't be a universal covering. The theory of universal covering is a theory of pointed spaces: A pointed cover (Y,y) of a pointed space (X,x) is a universal covering if, for any pointed cover (Z,z) of (X,x), there is a unique X-morphism $f:(Y,y)\rightarrow (Z,z)$ of pointed covers. | |
| Nov 14, 2010 at 23:46 | comment | added | ACL | @Dmitri: All of this is a matter of convention. Many theorems apply to non-empty connected sets (think of the union of two connected sets whose intersection is connected...). The empty manifold is the coproduct of the empty family... Higher homotopy suggests that it is reasonable not to view the empty set as connected. Anyway, homotopy theory is essentially a theory of pointed spaces, and pointed spaces are not empty. | |
| Nov 14, 2010 at 22:49 | comment | added | Andreas Blass | I would define a space X to be connected if, whenever it is expressed as a topological disjoint sum (i.e., coproduct) of spaces, one of the summands is X itself. This leads to the conclusion that the empty space is not connected, because t is the topological disjoint sum of zero spaces. (Analogously, I would define a positive integer to be prime if, whenever it is expressed as a product, it equals one of the factors. That makes the empty product, 1, not prime.) | |
| Nov 14, 2010 at 17:24 | comment | added | Dmitri Pavlov | @Georges and ACL: No, the empty space is by no means connected, because many theorems containing the word “connected” will fail if it is. For example, every manifold is the coproduct of a unique family of connected manifolds. A space is connected (in the appropriate sense) if and only if its π_0 is a one-element set. The π_0 of the empty space is empty. | |
| Nov 14, 2010 at 14:27 | comment | added | Georges Elencwajg | Dear ACL, I agree with you on the connectedness questions (Scott converted me on the zero number of components). But I still do not agree with simple-connectedness : if the empty space were simply-connected, wouldn't it be the universal covering of every space ? | |
| Nov 14, 2010 at 14:00 | comment | added | ACL | The empty space IS connected, and (but) has zero connected components: the connected components are the equivalence classes of the relation on the set defined by lying in a common connected subset. The partition in connected components is the empty partition of the empty set, as in the question leading to this thread. It is also simply connected in the sense of the theory of coverings: all of its coverings are trivial. | |
| Nov 14, 2010 at 11:55 | comment | added | Georges Elencwajg | Dear Scott: I thought of zero connected components just after posting (as usual!) and your excellent arguments convince me that this is the better point of view . | |
| Nov 14, 2010 at 10:22 | comment | added | S. Carnahan♦ | If we assume disjoint union is additive on the number of connected components, then the empty space should have zero connected components. I think this should correspond to the usual distinction between primes (connected spaces) and units (empty spaces) in situations where we have unique factorization. | |
| Nov 14, 2010 at 9:57 | comment | added | Georges Elencwajg | Dear Tom, it has one component, like all connected spaces. Bourbaki (and many textbooks) consider that the empty space is connected; but on the other hand a mathematician for whom I have enormous affection and admiration thinks, like you, that the empty set is not connected. So I suggest peaceful coexistence between the two points of view. Or maybe we should adopt the motto "don't ask, don't tell" on this question of connectedness? | |
| Nov 14, 2010 at 3:46 | comment | added | Tom Goodwillie | Agreed, except that I strongly prefer not to call the empty space connected. How many components does it have? | |
| Nov 14, 2010 at 3:28 | history | answered | Georges Elencwajg | CC BY-SA 2.5 |