TITLE: Various definitions of "topological immersion"
QUESTION [5 upvotes]: In Spivak's book on differential geometry he defines a topological immersion $f$ as "$f$ is a continuous function that is locally one-one". In my limited experience with the category $\mathsf{Top}$, this seems like the wrong definition, and I was wondering if anyone here agreed.
Wouldn't the definition consistent with the $\mathsf{SM}$ definition be "$f$ is a local homeomorphism onto its image"? It seems strange to think that for instance, every injective map into an indiscrete space falls under the heading of immersion.

REPLY [2 votes]: Surely you mean "$f$ is locally (in the domain) an embedding" rather than homeomorphism. In any case, the goodness or badness of a definition depends precisely on what he wants to use it for. What does Spivak want to use this notion of 'topological immersion' for? Note that for topological manifolds, 'local embedding' and 'locally injective' are the same! I doubt he had non-Hausdorff spaces in mind here.
This notion of immersion has historical significance, in the case that the domain and codomain are both manifolds. See for instance Hirsch, "On piecewise linear immersions".