TITLE: What is the difference between CW-complex and Cellular complex?
QUESTION [6 upvotes]: Is every CW-complex is a Cellular space? Is its converse true? 
If it is true then what is the difference between them?

We include the definition of CW-complex in algebraic topology given by Whitehed in 1949:
Definition. A CW complex is a Hausdorff space $X$ together with a partition of $X$ into open cells (varying dimension) that satisfies two additional properties:

For each $n$-dimensional open cell $C$ in the partition of $X$, there exists a continuous map $f$ from the $n$-dimensional closed ball to $X$ such that

the restriction of $f$ to the interior of the closed ball is a homeomorphism onto the cell $C$, and
the image of the boundary of the closed ball is contained in the union of a finite number of elements of the partition, each having cell dimension less than $n$.  

A subset of $X$ is closed if and only if it meets the closure of each cell in a closed set.


Definition. A cellular space is a topological space $X$, with a sequence of subspaces 
 $$X^0\subset X^1\subset X^2\subset \cdots \subset X,$$
 such that $X=\bigcup\limits_{n=0} X^n$, with the following properties:

CS(1) $X^0$ is a discrete space.
CS(2)  for each positive integer $n$, there is an index set $A_n$, and  continuous map $\psi_i^n: S^{n-1} \to X^{n-1}$ for each $i\in A_n$ and disjoint copies $D^n_i$ of $D^n$ (one for each $i\in A$) by identifying the points $x$ and $\psi_i^n(x)$ for each $x\in S_i^{n-1}$ and each $i\in A_n$.
CS(3) A subset $Y$ of $X$ is closed iff $Y\cap X^n$ is closed in $X^n,$ for each $n\geq 0$.

REPLY [3 votes]: Both concepts as defined in the question agree. This is proved in the appendix of Hatcher's book "Algebraic Topology". See Proposition A.2.
See also the references given by Jeanne Lefevre.