On Pg. 103 of Hatcher's Algebraic Topology the author has defined what a $\Delta$-complex is:
This is what I have gathered from what the author writes:
A $\Delta$-complex is a collection of oriented (geometric) simplices $\mathcal S=\{\Delta_\alpha\}_{\alpha\in J}$ indexed by a set $\alpha$ along with a collection of sets $\{\mathcal F_\beta\}_{\beta\in K}$, where each $\mathcal F_\beta$ is a collection of faces of the simplices in $\mathcal S$ of a common dimension.
This data is purely combinatorial. Given a $\Delta$-complex $\mathcal S=\{\Delta_\alpha\}_{\alpha\in J}$ and $\{\mathcal F_\beta\}_{\beta\in K}$, we may form the quotient space $\bigsqcup_{\alpha\in J} \Delta_{\alpha}/\sim$, where sim identifies the faces in each $\mathcal F_\beta$ via the canonical isomorphism (using the orientation of the faces) between the faces in it.
Is the above correct? Or have I got the definition wrong?
Here is another trouble:
On the next page, Hatcher writes the following:
Since the face identifications that produce a $\Delta$-complex $X$ always preserve the orderings of the vertices, these identifications never result in two distinct points in the interior of a face getting identifies in $X$. (I understand this).
This means that $X$, as a set, is the disjoint union of a collection of open simplices --- simplices with all their proper faces deleted.
Each such open simplex $e^n_\alpha$ of dimension $n$ comes equipped with a canonical map $\sigma_\alpha:\Delta^n\to X$ restricting to a homeomorphism from the interior of $\Delta^n$ onto $e^n_\alpha$.
Namely, the closure of $e^n_\alpha$ is the quotient of one of the simplices from which $X$ was constructed, or a face of one of these simplices, and $\sigma_\alpha$ is the quotient map from this simplex or face to $X$.
I do not understand what the author is trying to say here. I understand the parts in italics to some extent while the part in bold went completely tangential.
Can somebody write an easier to read definition of a $\Delta$-complex. What is the idea we are trying to capture?