Timeline for What does the 3rd axiom of topologies defined by neighbourhood mean?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 10, 2014 at 0:26 | vote | accept | fyusuf-a | ||
| May 5, 2014 at 20:28 | comment | added | Todd Trimble | Well, admittedly I hadn't tried hard to condense it down into something more reader-friendly; I was jotting down some observations with the hope of maybe coming back to it later. Although this relational beta-module viewpoint on topological spaces is not a bad thing to contemplate, when you find the time and interest... | |
| May 5, 2014 at 20:19 | comment | added | fyusuf-a | I am sorry, but I would really want to build an intuition on topological spaces (which, I admit, was not clear from the first formulation of my question). Your argument is very interesting and shows a real erudition, but it seems rather involved. | |
| May 5, 2014 at 13:51 | vote | accept | fyusuf-a | ||
| May 5, 2014 at 20:17 | |||||
| May 5, 2014 at 10:53 | comment | added | Todd Trimble | (Cont.) Thus pseudotopological spaces are sets $X$ equipped with a relation $\xi: \beta X \to X$ satisfying a unit condition. What precisely carves out topological spaces among pseudotopological spaces is the imposition of an extra associativity or transitivity condition on $\xi$, a kind of lax version of the associativity condition for algebras over a monad, as explained in the relational beta modules article at the nLab. I submit that the third axiom of the OP probably bears comparison with that transitivity condition, but it would take some time to explain that carefully. | |
| May 5, 2014 at 10:48 | comment | added | Todd Trimble | Of course, this answer doesn't explain yet what's up with that third axiom. I'd be inclined to compare it to the characterization of topological spaces among pseudotopological spaces along the lines of ncatlab.org/nlab/show/relational+beta-module. To be brief: axiom 2 of the OP is compared to the condition that the principal ultrafilter at a point $x$ converges to $x$. Regarding ultrafilter convergence as a morphism $\xi: \beta X \to X$ in the (bi)category of relations $Rel$, this becomes a unit condition on $\xi$ relative to a ultrafilter monad structure $\beta$ on $Rel$. (Cont.) | |
| May 5, 2014 at 2:20 | history | answered | Todd Trimble | CC BY-SA 3.0 |