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 $K$.