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?"

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    $\begingroup$ I don't know enough to give you a good answer, but two comments from me. Firstly, I really like this question! Even if the answer turns out to be silly (I suspect the answer is "no", for a whole bunch of reasons), it's a great question to learn from. Secondly: if $X$ is a topological space (about which, a priori, you know nothing), and you know there exists a cell decomposition $(X, \Gamma)$ ($\Gamma$ as you defined it), that tells you something about the space $X$. It might be something you already know in this case (e.g. Hausdorffness), but that's proof of concept: it can tell you things! $\endgroup$ – Billy Feb 25 '18 at 19:22
  • $\begingroup$ Although this was not the initial definition, the topology on a CW-complex on a space $X$ is specifically designed to enable the construction of continuous functions $f:X \to Y$ by induction on the skeleta, and then by the definition of $f$ on each cell. This also enabled the construction of homotopies of such functions. The aim was also to get away from the use of simplicial complexes, which involve many more cells. The idea had a long gestation; in JHCW's papers submitted prewar, there was the notion of "membrane complex"! $\endgroup$ – Ronnie Brown Feb 25 '18 at 22:04

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).

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  • $\begingroup$ I'm slightly confused now, because on page 131 in Lee's book he mentions that "For finite complexes (which are adequate for most of our purposes), the definitions we have given so far serve well. But infinite complexes are also useful in many circumstances, and for infinite complexes to be well behaved, two more restrictions must be added", the two restrictions he means are obviously the C and W properties needed to be satisfied by the cell complexes. $\endgroup$ – Perturbative Feb 26 '18 at 16:20
  • $\begingroup$ However we've both arrived at the conclusion that cell decomposition as defined by Lee are not useful, so what does he mean when he says the definitions given so far serve well? $\endgroup$ – Perturbative Feb 26 '18 at 16:21
  • $\begingroup$ The C and W axioms are actually automatically true for any finite complex (the C axiom is trivial, and the W axiom follows from the fact that a continuous surjection from a compact space to a Hausdorff space is a quotient map). $\endgroup$ – Eric Wofsey Feb 26 '18 at 17:35

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.

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  • $\begingroup$ I am not building the space up inductively as you are assuming $\endgroup$ – Perturbative Feb 25 '18 at 19:25
  • $\begingroup$ Alright, maybe then I have the wrong definition of Cell complex. I'll see whether I'll find the definition of Lee $\endgroup$ – lush Feb 25 '18 at 19:29

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