Number of connected components of this complement

Question

Let $X$ be a locally finite simplicial complex and let $K$ be a finite subcomplex of $X$. Why is the number of connected components of the complement $X-K$ finite?

Answer 1

Let's assume that $X$ is connected or, equivalently, path connected.
Assume that all simplices of $K$ of dimension $<n$, and maybe some simplices of dimension $n$, but no simplices of larger dimension have been removed so far. Let $\alpha\in K$ be an $n$- simplex. We've already removed its proper faces from $X$, so only the open simplex $\text{int}(α)$ is left, and its open star $\text{st}(α)$, the union of the interiors of all supersimplices of $α$, is a connected neighborhood of $α$. Since there are only finitely many supersimplices of $α$, and each is path connected, $\text{st}(α)-α$ splits into finitely many path components. It follows that also $P_α-α$ has only a finite number of path components, where $P_α$ is the path component with $α$ of $X$ minus all simplices removed so far.
Since we perform only finitely many such steps, and $X$ is connected, we get a finite number of components.