I come from combinatorics and my notions of algebraic topology are very limited. I have a purely combinatorial definition of a certain set of "cells" and I want to know if what I have is "enough" to say that it is a CW-complex or if I need to prove more.

 * My cells of dimension 0 are just a discrete finite set.
 * My cells of dimension 1 are edges between those 0-cells (so basically, I have a graph)
 * My cells of dimension $n$ are given by **certain** subsets (not all) of 0-cells (the dimension here is a certain combinatorial property of this subset) 

A face $f_1$ is included in face $f_2$ if the subset of 0-cells corresponding to $f_1$ is included in the subset of $f_2$

With the following properties:

A face of dimension $n$ is always a a certain subset of faces of dimension $n-1$ (so my poset of face inclusion is graded) 

If $f_1$ is a face of dimension $n_1$ and $f_2$ a face of dimension $n_2$, if I intersect $f_1$ and $f_2$ (as sets of 0-cells), I get a face $f_3$ of dimension $n_3 \leq min(n_1,n_2)$ such that $n_3 = n_1$ iif $f_1 \subseteq f_2$ (and $n_3 = n_2$ iif $f_2 \subseteq f_1$).

In particular, if $f_1$ and $f_2$ are distinct and of similar dimension, their intersection will always be of dimension strictly smaller. 

I am quite convinced that this "complex" is a CW-complex and I'm actually in the process of proving that it is a polytopal complex (each cell can be realized as a polytope). 

My question is more of nomenclature. These properties are quite strong already, do they give a CW-complex? If not, what is missing? (And what is this thing called?)