/sci/ - Science & Math

Good morning!

>>11587301
Since you have the path concatenation operation, you could indeed modulo out the reparametrisation by taking the homotopy classes of the paths instead? That would still detect holes inside the path component as two paths going around it from different sides would not be homotopic to one another, and path components are pretty much what you would get out of it. For sure, you can recover the set of path components by doing this trick. Your claim about the subbasis is correct (since every path goes through at least one point and so the union of all your [math]U_x[/math]-sets is the whole path space. It could be simplified, though. If you instead set [math]V_x = \{ \gamma\colon I\to X\ |\ \gamma(0) = x\}[/math], you will also get a subbasis by the same argument, but these topologies need not coincide with the usual one (maybe worth checking, but I doubt it). Would they generate the same topology? If you have free time, please check it out.

Just a few quick observations. Have fun, anon!