A cell is any subset of the plane homeomorphic to a disk.
Could someone provide a complete proof that a triangle is homeomorphic to a disk? I have two ideas but I can't seem make them fully rigorous.
One would be via this explicit mapping $f$: Map the boundary of the triangle to the boundary of the disk, and since they are both Jordan curves we're okay so far. Now, map the centroid $C$ (? I think the centroid is always inside the triangle, unlike the orthocenter or circumcenter) of the triangle to the center of the disk and then map radii to radii. Specifically, for each point $P$ on the boundary of the triangle, map the line $PC$ to the line $f(C)f(P)$.
The other way would be showing something like all closed connected subsets of the plane with Euler characteristic 2 are homeomorphic.
Feel free to use any definition of homeomorphism or continuity that you would like.