I want to verify whether the arguments for the statement is true. I say that an abstract simplicial complex, $X$, is locally finite if $\deg(v)<\infty$ for all $v\in X(0)$, where
$$ \deg(v)=\vert \{ e\in X(1): v\in e \} \vert. $$
A simplicial complex is connected if its $1$-skeleton is connected, i.e, if it's connected as a graph. Assume $X$ is connected, i.e., its $1$-skeleton is connected as a graph.
The connected component of a locally finite graph is countable, as verified here, so the set of vertices in $X$ is countable. Since the degrees are all finite, we have no infinite simplices, and thus $X$ is a sub collection of the finite subsets of $X$. The collection of finite subsets of a countable sets is countable as a countable union of countable sets, and hence $\vert X\vert \leq \aleph_0$.
I want to verify this argument, to know whether under these relatively mild conditions, my complexes would then be nice and simple, and alot of pathologies are avoided.