Show that every simplicial complex is a CW-complex. Klaus Janich's Topology, Page 97:

The decomposition of a simplicial complex into its open simplices is a CW-decomposition.

No further explanation of this proposition is provided after that,so I'm trying to work it out.
There's not much difficulty in proving the axioms of "characteristic maps" and "closure finiteness", yet I have no idea about dealing with the axiom of "weak topology", i.e. $A\subset X$ is closed if and only if every $A\cap \bar e$ is.($e$ is a cell of the decomposition).
Any tips please?
The definition of simplicial complex in Janich's book:

Definition(Simplicial Complex or Polyhedron).A set $K$ of simplices in $R^n$ is called a simplicial complex or a polyhedron if the following three conditions are satisfied:
  
  
*
  
*If $K$ contains a simplex it contains all faces of this simplex.
  
*The intersection of two simplices of $K$ is either empty or a common face.
  
*(In case $K$ is infinite), $K$ is locally finite, i.e. every point of $R^n$ has a neighborhood that intersects only finitely many simplices of $K$.
  

 A: The key to this is the local finiteness condition. 
Suppose $x_i$ is a sequence in $A$ that converges to a point $p \in X$. By local finiteness, we may choose an open neighborhood $U$ of $p$ which intersects only finitely many closed cells $\bar e_1,\ldots,\bar e_K$. Then we may choose $I$ so that if $i \ge I$ then $x_i \in U$ and therefore $x_i \in \bar e_1 \cup\cdots\cup \bar e_K$. Since there are infinitely many $i>I$ but only finitely many $k=1,\ldots,K$, there exists an infinite subsequence $i_1<i_2<i_3<\cdots$ and a $k \in \{1,\ldots,K\}$ such that $x_{i_j} \in \bar e_k$ for all $j$. Since the limit of a sequence equals the limit of any subsequence, it follows that $p \in \bar e_k$. But $x_{i_j} \in A \cap \bar e_k$ which is a closed subset of $\bar e_k$, so $p \in A \cap \bar e_k \subset A$.
A: Any closed simplex in $\mathbb R^n$ is closed subset, so it will be closed subset in $X\subset \mathbb R^n$. So desired statement becomes tautological, because induced topology on $\Delta^k$ from $\mathbb R^n$ coincides with induced one from $X$.
In fact, both $CW$-complexes and simplicial complexes has factor-topology, obtained by gluing $k$-cells/simplexes to the $(k-1)$-skeleton.
