A path which its domain is a general compact set.

[MathematicalPhysicist]

How is a general path called instead of being a continuous function from an interval to some topological space, where we replace the domain from an interval to a compact set, is there a name for such a function?

Perhaps I should add that the compact set is also convex.

 

[WWGD]

But a path is usually defined as a continuous map $f: I \rightarrow X$, where $I=[0,1]$, which is compact and convex. Of course, you may deal with open, half-open intervals too.

 

[MathematicalPhysicist]

I want to somehow to generalize the notion of a path.

 

[HallsofIvy]

Generalize it in what way? That is, what properties of a "path" do you want to keep?

 

[MathematicalPhysicist]

@HallsofIvy That it connects two points on the range which aren't necessarily the same and that the range is compact and convex. Such as if the domain is S^2, then the north pole is mapped to some point in X and the south pole to some point in X, and in case they are mapped to the same point, it's the same as in the case of path homotopy, where you define the first fundamental homotopy; Is there a name for such construction?

I can see that it's called in case of S^n and [0,1]^n as n-homotopy paths, what happens in case the domain is only compact, and not necessarily convex, is such a construction still called a homotopy?

 

[lavinia]

@HallsofIvy That it connects two points on the range which aren't necessarily the same and that the range is compact and convex. Such as if the domain is S^2, then the north pole is mapped to some point in X and the south pole to some point in X, and in case they are mapped to the same point, it's the same as in the case of path homotopy, where you define the first fundamental homotopy; Is there a name for such construction?

I can see that it's called in case of S^n and [0,1]^n as n-homotopy paths, what happens in case the domain is only compact, and not necessarily convex, is such a construction still called a homotopy?

While I am not totally sure, homotopy to me always refers to a path of maps. Two functions, f and g, from a space, X into a space Y, are homotopic if there is a map from XxI -> Y such that F(x,0) = f(x) and F(x,1) = g(x). The space X need not be compact.

In the case of the sphere, one has a surjective map from IxI -> S^2 which crushes the top of the square to the north pole and the bottom to the south pole and maps the vertical lines (a,t) to great circles. So this reduces to the case of a homotopy between two constant paths.

 

[WWGD]

AFAIK, you can define your homotopies using different indexing sets, not necessarily the unit interval, if that is what you were asking. Of course, if you want a continuous deformation, you need a continuous index, which I think is equivalent to having a dense ordering, i.e., a relation < so that if x<y , there is z so that x<z<y.