TITLE: Homeomorphism definition
QUESTION [5 upvotes]: I was told by my professor that homeomorphisms are continuous maps with continuous inverse, but do those conditions also imply that the map is bijective?

REPLY [3 votes]: It was inconvenient that your professor worded it this way. From the get-go, saying that a homeomorphism is continuous function with a continuous inverse assumes that we have some function $f$, and that its inverse, $f^{-1}$, exists. Off-the-bat, just cause we have a function $f$, it does not mean that it has an inverse. 
In addition, the way your professor phrased it doesn't make it clear whether we are talking about a subset of the range, or the whole range itself: in a homeomorphism, we must have the whole range, which is something your professors phrasing neglected to capture. 
If I were you, I would just forget about what your professor said. It is kind of circular. Just to make it clear. By definition, a homeomorphism is a function $h$, from a topological space $X$, to a topological space $Y$, such that the following hold: 

$h$ is 1-1 
$h$ is onto
$h$ is continuous
$h^{-1}$ is continuous

Note, we do not say $h^{-1}$ exists here: this is a consequence of $h$ being 1-1, as we can always create $h^{-1}$ in such a case. ALSO, we imply that the image of $h$ is all of its range: this is captured by saying that $h$ is a bijection.