Examples of finite simplicial sets

Question

Let $K$ be a simplicial set. A simplex $x\in K_{n}$ is said to be non degenerate if it is not the degenerancy of a $n-1$ simplex, i.e if there is no $y\in K_{n-1}$ such that $s_{i}y=x$. A simplicial set is said to be finite if it contains a finite number of non degenerate simplices. How I prove that a simplicial set is finite? can I deduce that by its topological realization? Are for examples $\Delta[n]$, $\partial\Delta[n]$ and the horns ${\Lambda_{n}}ˆ{k}$ finite simplicial sets?

Answer 1

The realization of a finite simplicial set is compact, as all degenerate simplices get collapsed down into the non-degenerate (nice) simplices of which they are degeneracies.
Conversely, if the realization of a simplicial set $X$ is compact, then there can only be a finite number of nice simplices. To see this, let $C\subseteq|X|$ be compact. Now for each nice $n$-simplex $x$, choose a point $p_x$ in $C\cap\mathring\Delta^n_x$ whenever possible, where $\mathring\Delta^n_x$ means the image of the interior of the standard-$n$-simplex under the map $\Delta^n\to|X|$ corresponding to $x$. The resulting set $P$ is closed and discrete since its preimage under every map $\Delta_y^k\to|X|$ for every simplex $y\in X$ is finite. Hence $P$ must be finite. That implies $C$ meets only finitely many interiors of nice simplices. In particular, when $C=|X|$, then $X$ is finite.