What are some uses of non-locally finite CW complexes? A CW complex can be non-locally finite. An example of one is obtained by attaching a disk by its boundary to each point of $(0, 1)$, together with the end-vertices $\{0\}$ and $\{1\}$. When proving theorems, non-locally finite CW complexes cause additional difficulties. For example, the topology in a product CW complex in general is not the product topology, but its k-ification (which is a finer topology). One way to fix this is to require one of the factors to be locally finite, which makes the product topology compactly generated (i.e. the product topology is its k-ification). In general, locally finite CW complexes seem to behave well under various operations.
What are some uses of non-locally finite CW complexes? Are they important in some area of mathematics? What would be lost if CW complexes were defined to be locally finite to begin with?
 A: An enormous number of naturally occurring examples used in homotopy theory are non-locally finite CW complexes.  Among the most basic examples are the infinite dimensional projective spaces $\mathbb{RP}^\infty$ and $\mathbb{CP}^\infty$, or more generally infinite-dimensional Grassmannians and other classifying spaces of groups.  There are also a lot of constructions that produce spaces (sometimes referred to as "designer homotopy types") that are not particularly geometrically natural but are extremely useful for the formal algebraic properties that their invariants such as homology and homotopy groups have which enable computations about more natural spaces.  Often they involve infinite iterative constructions that repeatedly adjoin cells to produce some crazy huge CW complex that is not locally finite.  Examples include Postnikov and Whitehead towers, Eilenberg-MacLane spaces, and spaces produced by the Brown representability theorem.
I would also mention that algebraic topologists almost always prefer working in something like the category of $k$-spaces anyways, and so for instance the difficulties with products of non-locally finite CW complexes are irrelevant.  This is not just in order to make non-locally finite CW complexes better behaved--instead, the primary motivation is to have a well-behaved theory of mapping spaces.
