Rotman: Examples of identifications that are not homeomorphisms

Question

I'm working through Rotman's Algebraic Topology, and he defines an identification as a map this is continuous and an open mapping, i.e. $f: X \to Y$ is an identification if $U$ is open in $Y$ if and only if $f^{-1}(U)$ is open in $X$. He then goes on to prove corollary 1.10 that gives a way of constructing a homeomorphism from an identification.

Identifications seem like such nice maps to me that I can't think of a good example of one that is not a homeomorphism. Having such an example would help make this corollary more concrete.

Answer 1

Note that what Rotman is calling an identification is often called a quotient map, which are maps that we typically think of as "identifying" points of some space to get a new space. Other examples of quotient maps that are not homeomorphisms are

1. The quotient map $\Bbb R^{n+1}\smallsetminus\{0\}\to\Bbb R\Bbb P^n$, where $\Bbb R\Bbb P^n$ is real projective $n$-space. This isn't a homeomorphism because the projective space is actually compact, while the punctured Euclidean space is not.

2. The quotient map that identifies corresponding points in two copies of $\Bbb R$ except for their origins. The resulting space is known as the line with two origins and it isn't Hausdorff, though the domain is Hausdorff.

3. The quotient space where we identify two points $\alpha,\beta\in\Bbb R$ if and only if $\alpha-\beta$ is an integer multiple of $2\pi$ is homeomorphic to the circle, but it isn't homeomorphic to $\Bbb R$ because again the circle is compact while $\Bbb R$ is not.

4. The quotient map that identifies all points in a given space. The resulting space is a singleton, which is seldom homeomorphic to the original space.

Answer 2

Never mind, by thinking a bit more carefully, I have an idea: $I \times I$ is the unit square, $h$ is an identification that takes the square to a cylinder $C$ by mapping both the top and bottom edges to the same part of the cylinder, which causes $h$ to not be a bijection. By modding these two edges together in the quotient space we then get a homeomorphism from $I \times I \: / \sim \to C$ simply because we made $h$ bijective.