What is meant by gluing two metric spaces together?
To perform the gluing, we need:
- two metric spaces $X$ and $Y$
- a set $A\subset X$, this is the part of $X$ covered in glue
- an isometric embedding $f:A\to Y$, which is a way to put the glue-covered part of $X$ over $Y$.
After we firmly press the spaces together and let them sit for a while, a point $x\in A$ becomes identical with the point $f(x)\in Y$. The resulting space can be described as the quotient $(X\sqcup Y)/(x\sim f(x))$. It is usually denoted $X\cup_A Y$. Its metric is the standard quotient metric; however, in this case the formula for quotient metric can be simplified to $$ \tilde d(p,q) = \begin{cases} d_X(p, q) & \mbox{if } p, q \in X \\ d_Y(p, q) & \mbox{if } p, q \in Y \\ \inf_{a\in A} [d_X(p, a)+d_Y(q, f(a))] & \mbox{if } p \in X \mbox{ and } q \in Y \\ \inf_{a\in A} [d_Y(p, f(a)) + d_X(q, a)] & \mbox{if } p \in Y \mbox{ and } q \in X \end{cases} $$
A simple example to start with: $X=(-\infty,0]$, $A=\{0\}$, Y=$[0,\infty)$, $f(0)=0$. This means we glue two half-lines together at point $0$. The result is $\mathbb R$ with the standard metric.
Another example: glue two closed disks of the same radius along their boundaries. This means $A$ is the boundary of one disk, and $f(A)$ is the boundary of the other. The resulting space is homeomorphic to a sphere, though the metric on each disk is still flat. It's like a sphere that someone sat on.