I have been thinking about methods for constructing continuous paths locally in a space. These paths have domain the unit interval and map into "small" neighborhoods of points in a space. Moreover their images are constrained in ways which are unimportant for this question.
If the space is first countable (points have countable bases for the topology) and locally path connected (lpc) then the desired constrained paths are easily constructed.
Here is my question.
Are the spaces which arise naturally in stable homotopy theory (e.g. infinite loop spaces, spaces arising in "brave new algebra") locally nice in this sense (i.e. first countable and lpc)?
If not, what are some natural examples which fail either to be first countable or locally path connected (or both)?
Edit: My failure to offer more motivation for my question has led the discussion off in an unexpected (to me) direction. I now see that the question itself may be wrong-headed. Still, let me fill in the background which led to my question.
I am not motivated here by any homotopy theoretic considerations per se. Instead, I am studying the properties of functors from categories of point-set topological spaces to categories like CAT, the category of small categories and functors.
The simplest example is the Moore-path functor. This assigns to a space the small category whose objects are the points of the space and whose morphisms are the parametrized continuous paths in the space. I am wondering on just what subcategories of spaces is this functor full (it is trivially faithful).
I think one can show that the Moore-path functor is full on the subcategory of spaces whose objects are first countable and locally path connected.
My question then comes down to asking whether or not there is a class of "natural models" of stable homotopy types which lives in this category or whether instead weaker local hypotheses on the spaces are called for.
My actual interest involves related higher categorical notions but this seemed like a good place to start.
Thanks to everyone who has tried to illuminate the situation for me.