Timeline for Defining topological spaces with the notion of continuous path
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 30, 2012 at 20:15 | answer | added | Anton Fetisov | timeline score: 1 | |
| Jan 30, 2012 at 18:26 | answer | added | Ronnie Brown | timeline score: 1 | |
| Jan 27, 2012 at 16:34 | comment | added | Guillaume Brunerie | @Giorgio Not exactly. I want to know what are the conditions on a set X and a collection of maps $I^n\to{}X$ such that these maps are exactly the continuous maps for some topology on X. | |
| Jan 27, 2012 at 15:57 | comment | added | Andrew Stacey | I think I've heard of this as arc generated topology. It certainly isn't a definition of all topological spaces, but is a subcategory (a fairly nice one too). As Mike says, it's usual to have a little more than just the arcs. | |
| Jan 27, 2012 at 15:38 | comment | added | Giorgio Mossa | @GuillaumeBrunerie you're asking if knowing what are the continuous functions of type $I^n \to X$ characterize the topology, am I right? | |
| Jan 27, 2012 at 15:30 | answer | added | Mike Shulman | timeline score: 3 | |
| Jan 27, 2012 at 12:12 | comment | added | Guillaume Brunerie | @Ricky Yes, composition is concatenation. | |
| Jan 27, 2012 at 6:32 | comment | added | Matt Brin | Strange things can be done with this concept. I cannot find the reference in Math Reviews, but I recall a paper where the author put a topology on the plane so that the only continuous paths were piecewise linear. | |
| Jan 27, 2012 at 5:49 | comment | added | Mike Shulman | I think concatenation of continuous paths is commonly called "composition" by category theorists and homotopy theorists, thinking of paths as like morphisms in a category of points. | |
| Jan 27, 2012 at 4:13 | comment | added | user5810 | Do you mean "joining of two continuous paths is continuous"? $\hspace{2 in}$ Composition of paths would in general not make sense. $\;\;$ | |
| Jan 27, 2012 at 0:42 | history | asked | Guillaume Brunerie | CC BY-SA 3.0 |