In Stillwell's Geometry of Surfaces, he defines a locally euclidean surface $S$ as a set equipped with a distance function $d_S(A,B)$ defined for every $A,B \in S$ such that for every $A \in S$, there exists an $\epsilon > 0$ such that every $\epsilon$ neighborhood of $A$ is isometric to a euclidean disc.
He then defines: a euclidean surface $S$ is connected if each $A, B \in S$ are connected by a polygonal path $\Sigma$ (never defines polygonal path); whose sides lie within euclidean discs of $S$.
Lastly he defines: On a euclidean surface $S$ we can define a line segment to be a polygonal path whose successive sides (each lying within a euclidean disc of $S$ where the notion of "line segment" is already meaningful) meet at angle $\pi$. Then $S$ is called complete if any line segment can be continued indefinitely.
He says that the punctured plane is not complete and then asks if several surfaces are 1) locally euclidean 2) connected and 3) complete.
The first is the example of the union of the planes $x = 0$ and $y=0$. I believe this is not euclidean, because at the intersection of the two planes there can't be an isometry between points on the different planes, but I could be wrong.
The other examples include: the surface of an infinite triangular prism, the surface of a cube, the surface of a cube with the vertices removed, a closed disc, and an open disc.
The definitions are quite informal and I think that is tripping me up, along with the fact that I'm self studying. Any insight on how to better understand these definitions would be appreciated.