I find the following definitions in a book about algebraic topology:
Definition: Let $K$ be an abstract simplicial complex.
$(1)$ If $K$ is finite, simply connected and with nonempty boundary, then we say $K$ is a combinatorial closed disk.
$(2)$ If $K$ is infinite, simply connected and without boundary, then we say $K$ is a combinatorial open disk.
Now, I understand that when we talk about a topological disk we refer to a topological space homeomorphic to a disk. What I would like to understand is why the simplicial complex needs to be finite or infinite in the above definitons.
As an example on $\mathbb{R}^2$, is there an infinite disk on $\mathbb{R}^2$ which is closed (think in terms of compactness)? Think intuitionistically on this one.