Fibre bundles as generalized universal covers

Classical covering theory ensures each connected metric space with reasonable local properties ( locally path connected, and semi locally simply connected) admits a universal covering (essentially uniquely).  In this case the preimages of the natural covering map are discrete.

How might one reasonably generalize the latter facts, sacrificing discreteness of fibres and nice local properties of the base, and obtain a fibre bundle rather than a traditional covering map, whose total space is simply connected, so that all paths in the base space lift uniquely, and so that all fibres are totally disconnected?

There is no universally agreed upon definition of generalized universal cover.

However we will discuss one of the most natural definitions, see why fibre bundles are often hopeless to obtain, but present a substantial class of nontrivial examples where fibre bundles are indeed achieved. along with the other mentioned properties.

In particular, certain planar sets generate bundles whose fibres are the classical free topological groups over countably many generators in the sense of Graev or Markov.

This talk is motivated by a question of Andrej Bauer.
 http://mathoverflow.net/questions/165684/a-generalization-of-covering-spaces-to-fiber-bundles-with-totally-path-disconnec [1]