10
$\begingroup$

The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful topological invariants? Things like compact, metric, complete and ordered all seem fairly important but it's easy to ignore where they're being relied upon (precisely because the definition of path is so intuitive).

The obvious follow up question would be, can we replace $I$ with some other space and still develop an interesting theory? I guess you could go either way, either weakening or strengthening the definition of path.

  • 0
    If you defined a path in $X$ as a continuous map from $\mathbb{R} \to X$, then how would you concatenate two such paths?2012-01-20
  • 1
    (somewhat) related: http://mathoverflow.net/questions/28380/a-canonical-and-categorical-construction-for-geometric-realization2012-01-20
  • 2
    This might be relevant: http://mathoverflow.net/questions/80777/what-is-a-continuous-path (I would like to make this a comment, but I don't have enough reputation.)2012-01-20
  • 0
    @wckronholm So you're saying we need the domain to have nontrivial boundary, like for the higher homotopy groups? Those two links are great!2012-01-20
  • 0
    @dls It seems like you would need some kind of boundedness condition, like $\gamma(t) = x_0$ for $t<<0$ and $\gamma(t) = x_1$ for $t>>0$. Then you could concatenate two paths, up to homotopy. (Now it's starting to feel like the difference between knots and long knots.)2012-01-20
  • 0
    @dls could you try to be more explicit? In particular what do you mean when you say "what properties of the interval give rise to useful topological invariants"?2012-01-20
  • 2
    It think that's about as precise as I can be for a soft question. Like I said, the two links to MO questions are exactly what I was looking for. Here's another example I found this morning (though I understand this one even less): http://en.wikipedia.org/wiki/A%C2%B9_homotopy_theory2012-01-20
  • 0
    There is a generalization of a path, called a Moore path, where we essentially do what @wckronholm suggests -- the space of Moore paths tends to be homotopy equivalent to the space of honest paths, so these kinds of paths don't really encode any new information for us, they are just easier to work with in some contexts.2015-10-07

1 Answers 1