Are all cell decompositions useful? Based on the definitions and conventions used in the book Introduction to Topological Manifolds by John Lee, I can do the following.
Let $X$ be a Hausdorff space, then define $\Gamma = \{\{x\} \ | \ x \in X\}$, that is $\Gamma$ is the collection of all singletons of $X$, and is trivially a partition of $X$ and thus makes $(X, \Gamma)$ a cell-complex. We don't even need to specify any characteristic maps since all the open cells in $\Gamma$ are of dimension $n = 0$.
This cell decomposition seems to be (on a superficial glance) somewhat useless, as it doesn't really shed any insight into anything about $X$ really. If we have such a trivial cell decomposition why are cell complexes even useful as a tool 'for telling more information about a space' in Topology?
I'm guessing that we need a nicer cell decomposition of $X$ for which we can extract meaningful information from $X$. If so what conditions do we need for a 'nicer' cell decomposition of $X$?
Finally this all leads me to the question: "Are all cell decompositions useful?"
 A: No, cell decompositions as defined by Lee are not useful in their full generality.  They are just an intermediate step in the definition of a CW-complex, which is the definition that is actually important and useful.  In fact, many people use the term "cell complex" to mean "CW-complex", rather than the much weaker notion that Lee defines.
If you restrict your attention to cell decompositions that have only finitely many cells, however, then every cell complex is actually automatically a CW-complex, and so the weaker definition is perfectly fine.  Historically, people were first interested in such finite cell complexes, and the only later came up with the correct way to generalize the definition for infinite complexes (to get the general definition of a CW-complex).
A: Your decomposition of X is not really a decomposition of X:
What you get by building X this way is X together with the discrete topology on it. You declare all the sets {x} to be open, which means that X will be discrete.
If you're not looking at a discrete space, the space you build happens to just be another topological structure on the same set. But it is not the original topological space.
So it is not true, that any space has a CW-Structure.
