Homeomorphisms

• When Is a Proper Injective Map a Homeomorphism onto Its Image?
  A continuous proper injective map into a first-countable Hausdorff space is an embedding; without Hausdorffness, the statement can fail.
• When Is a Covering Map a Homeomorphism?
  A covering map is a homeomorphism exactly when every fiber has one point; the same local-sheet argument compares two covering spaces.