Timeline for Continuous relations?
Current License: CC BY-SA 3.0
4 events
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| Aug 25, 2014 at 12:19 | comment | added | Joonas Ilmavirta | @JoelDavidHamkins, it depends on the setting, but true, functions are not the only relations with preferred direction. The way I'm used to looking at relations is a symmetric one, but I am admittedly no expert here. I feel that a complete answer to the OP's question should address how the newly defined continuity respects different constructions of new relations from old ones, and I attempted to discuss that side. But as I wrote, symmetry is not something we can hold on to if we want to generalize continuous functions. | |
| Aug 25, 2014 at 12:09 | comment | added | Joel David Hamkins | I don't agree with the comments in your first point. A relation $R$ on $X\times Y$ is a relation from $X$ to $Y$, with the "preferred direction" being from $X$ to $Y$. The preferred direction is used when defining the domain and range of a relation, just as for a function. Why should we expect or want that $R$ is continuous if and only if the inverse relation is continuous? As you mention, we don't generally have that for functions. Many relations have other properties not shared by their inverses. For example, if $R$ is well-founded, the inverse $R^{-1}$ is not generally well-founded. | |
| Aug 23, 2014 at 15:00 | comment | added | Lehs | I think the problem with the common definition of continuous functions is that it is optimized to functions (defined on the entire domain). Extending this definition to relations makes even the very smooth unit circle discontinuous, contrary to my intuition. Two objective conditions are that the definition should works for ordinary functions and that the composition of continuous relations (that extends the composition of functions) should be continuous. | |
| Aug 23, 2014 at 13:43 | history | answered | Joonas Ilmavirta | CC BY-SA 3.0 |