We do it all the time. An index is just indicative that there is a mapping (a function), usually from the integers. However we don't use the subscript notation

We do it all the time. An index is just indicative that there is a mapping (a function), usually from the integers. However we don't use the subscript notation when indexing by a continuum due to the discomfort you describe. The point is that we need some way to deal with objects that are inherently infinite-dimensional.