Timeline for Continuous relations?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 25, 2014 at 20:12 | comment | added | Lehs | @Eric Wofsey: is the composition of continuous relations continuous according to your definition? | |
| S Aug 25, 2014 at 19:56 | history | suggested | Lehs | CC BY-SA 3.0 |
made the definition easier to find and refer to
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| Aug 25, 2014 at 19:54 | review | Suggested edits | |||
| S Aug 25, 2014 at 19:56 | |||||
| Aug 24, 2014 at 10:33 | comment | added | Eric Wofsey | @JoonasIlmavirta: That definition will give you closed relations, which are not very well-behaved other than being symmetric in $X$ and $Y$. They are not closed under composition and include functions that are not continuous as functions. If you replace $K$ by $Y$ (so you are looking locally only on $X$), you get my definition of continuity (since it is local on the domain). | |
| Aug 24, 2014 at 10:06 | comment | added | Joonas Ilmavirta | @NikWeaver, if $X$ and $Y$ are locally compact Hausdorff, wouldn't it be a natural attempt to define that a relation $R\subset X\times Y$ is continuous if $R\cap(C\times K)$ is continuous for all compact $C\subset X$ and $K\subset Y$? | |
| Aug 23, 2014 at 22:17 | comment | added | Nik Weaver | A natural next question is the locally compact Hausdorff case ... | |
| Aug 23, 2014 at 20:48 | comment | added | Dimitri Chikhladze | "Homomorphic relations" or modules can be also defined between relational algebras, and that might coincide with your general definition of the continuous relation. | |
| Aug 23, 2014 at 20:46 | comment | added | Dimitri Chikhladze | General topological spaces can also fit into the algebraic picture. Topological spaces are relational algebras for the ultrafilter monad. For a relational algebra the structure map $TA \rightarrow A$ is a relation, i.e. morphisms of $Rel$. The algebra equations are replaced by inclusions of relations, i.e. 2-cells of $Rel$. This reflects the fact that in the non-compact Hausdorff case ultrafilters can converge to any number of points. | |
| Aug 23, 2014 at 18:28 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
added 5 characters in body
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| Aug 23, 2014 at 18:16 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
added 149 characters in body
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| Aug 23, 2014 at 17:32 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
previous edit was incorrect
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| Aug 23, 2014 at 17:20 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
added 14 characters in body
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| Aug 23, 2014 at 16:54 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
added 47 characters in body
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| Aug 23, 2014 at 16:46 | comment | added | Lehs | Limit needs Hausdorff, but has it to be an operator? If you use a relation instead, wouldn't then a general topology be generated from the relation: $x_\alpha C x \Leftrightarrow$ "$x$ is a condensation point of $x_\alpha$"? | |
| Aug 23, 2014 at 16:44 | history | edited | Eric Wofsey | CC BY-SA 3.0 |
added 47 characters in body
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| Aug 23, 2014 at 15:40 | history | answered | Eric Wofsey | CC BY-SA 3.0 |