CW-complexes are defined by attaching cell with *increasing dimension*: you start with a set of points, then attach 1-cells, then 2-cells and so on. Why are defined so? My question is: why is it necessary to attach cells ordered by dimension and not to attach a 2-cell, then a 1-cell, then a 3-cell...? The proofs I have seen about CW-complexes are mainly done inductively on the dimension of the complex, but I guess that they could be done also inductively on the number of cells attached (for example I am thinking about the uniqueness theorem for homology theories). Are there theorems that holds only for standard CW-complexes and not for these?



CW complexes obtained by attaching cells not with increasing dimension
There is a name for the kind of space you are describing: a **cell complex.**
A CW complex is a cell complex which has cell attachments in the increasing order of dimension.


The main advantage of having a CW complex over a mere cell complex is that the filtration by skeleta defines a finite chain complex model for its singular homology: if $X$ is a CW complex with $k$-skeleton $X^{(k)}$ then the homology of the pair $H\_\ast(X^{(k)},X^{(k-1)})$ is 
concentrated in degree $k$ and is free abelian in that degree. The composition
$$
H\_k(X^{(k)},X^{(k-1)}) \overset{\partial}\to H\_{k-1}(X^{(k-1)}) \to H\_{k-1}(X^{(k-1)},X^{(k-2)})
$$
defines the boundary operator for cellular homology.


Howeover, in the cell complex case, there is no such filtration giving rise to a finite chain complex. The best thing you can do is use the partial ordering of cell attachements to define a filtration indexed by a poset whose relative homology defines a spectral sequence. This is more cumbersome to work with.


There is a thirty year old work by Igusa and Waldhausen which studies **families** of cell complexes--defining a sort of moduli space called the **expansion space**. The work was never published. Roughly speaking, they prove that the moduli space of cell complexes which are pointed and contractible give a model for the $h$-cobordism space of a disk. 


