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further, codimenson 0 immersions do not behave like other immersions, which are largely determined by the stable normal bundle: in codimension 0 one has issues of fundamental class and cover spaces.
further, codimenson 0 immersions do not behave like other immersions, which are largely determined by the stable normal bundle: in codimension 0 one has issues of fundamental class and cover spaces.

a codimension 0 immersion of a closed manifold is precisely a covering map, i.e., a fiber bundle with 0-dimensional (discrete) fiber.
a codimension 0 immersion of a closed manifold is precisely a covering map, i.e., a fiber bundle with 0-dimensional (discrete) fiber.

=== codimension 0 ===codimension 0 immersions are equivalently "relative" dimension 0 "submersions," and are better thought of as submersions.
=== codimension 0 ===codimension 0 immersions are equivalently "relative" dimension 0 "submersions," and are better thought of as submersions.

another way of understanding this is that a codimension "k" immersion of a manifold corresponds to a codimension 0 immersion of a "k"-dimensional vector bundle, which is an "open" manifold if the codimension is greater than 0, but to a closed manifold in codimension 0 (if the original manifold is closed).
another way of understanding this is that a codimension "k" immersion of a manifold corresponds to a codimension 0 immersion of a "k"-dimensional vector bundle, which is an "open" manifold if the codimension is greater than 0, but to a closed manifold in codimension 0 (if the original manifold is closed).