Question: Consider the following construction: take any uncountable set of elements and form a chain ($x < y < z <\dots$ ) of the elements (in particular, I think one can well-order the reals using the AoC, and then take this ordering) and now consider these elements to be 0-cells. Now attach a 1-cell between two 0-cells $x, y$ if there is no $z$ between $x$ and $y$. [Intuitively, this kind of looks like a "really long" real line.] The claim is that this is not a CW-complex. But why?
Motivation: On a final, I was asked a question which sparked my interest in this finite vs infinite distinction in CW complexes. In particular, I thought that this "long line" construction, though somewhat artificial, should qualify as a "nice space" since it is contractible. When I asked about it, I was told that this was not a CW complex because the topology restricts us to putting a finite number of cells in each dimension. I'm not quite sure why this is true.
Can anyone shed some light on this for me?