In the turn-based strategy video game, Advance Wars, players are able to create their own game boards. The boards are 2D grids composed of discrete tiles with various "movement costs". Thinking of this as a graph, the "movement cost" of a tile represents the distance from all adjacent tiles.
However, a bug-turned-feature discovered by the game's PvP community's board designers is that a hacked/crafted null tile would be rendered as a black square, and have a movement cost of $0$. This was dubbed the "teleport tile" and used to various effects.
One such effect, depicted here, is to create a map for which all edge tiles (ocean tiles in this case) are distance-$1$ from each other, but is otherwise a conventional 2D "plane":
What would a topologist call this? I'm inclined to say it's a "sphere": with one pole being represented by the black edge, but I'm not educated on the topic enough to say. (No units may end their turn on the black squares, so you may consider them not to be genuinely "part of" the board if it would yield a more elegant answer. Their chief effect is that any tiles adjacent to a contiguous region of them become "adjacent" to one another.)
Best Answer
Yep, that's a sphere.
You can tell, because if you take a journey on the map that leads you back to the starting point, then you can shrink that journey down to a single point:
So, by the Poincare conjecture, it's a sphere: https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture