I'm self-studying "Elements of Algebraic Topology" by Munkres these days.
In that book, Exercise #2 (d) in section 15 asks you to prove the following
If $K$ is $n$-dimensional finite simplicial complex, (Lebesgue) covering dimension of $\vert K\vert$ is at most $n$.
I can prove that covering dimension of $n$-dimensional simplex is at most $n$. That is easy because you can think $n$-dimensional simplex as closed set in $\mathbb{R}^n$. However, I don't know how to approach to prove the simplicial complex case. Some clever gluing is necessary, maybe?
And here is follow-up question. In section 16, Exercise #3 asks you to show that any $n$-dimensional simplicial complex has at most $n$ covering dimension. So, number of simplexes in $K$ doesn't need to be finite in this more general problem.
I'm looking for hints or solutions to prove above two exercises. Thanks for any help in advance.