This is a question on nomenclature of $K$-theory in the topological category.

The $K$-theory of a compact space $X$ is defined as the Grothendieck group of the vectorbundles on $X$. The Atiyah-Jänich Theorem states that this is the same thing as the homotopy classes of maps $X\rightarrow \Phi(\mathbb{H})$, where $\Phi(\mathbb{H})$ is the space of Fredholm operators on some separable infinite-dimensional Hilbert space.

Now, for a non-compact space $Y$ the $K$ theory $K(Y)$ is not defined as the Grothendieck group of vector bundles on $Y$. One can do a couple of things:

- Define it as the $K$-theory of its one point compactification $K(Y):=K(Y_+)$. One needs to assume $Y$ is locally compact for this to make sense.
- Another option is to assume that $Y$ is nice, for example an infinite $CW$ complex such as $CP^\infty$, and to define the $K$-theory of $Y$ as the limit of its finite subcomplexes. Note that $CP^\infty$ is not locally compact.
- Yet another option is to define the $K$-theory of $Y$ as maps $Y\rightarrow \Phi(\mathbb{H})$. I believe this is called representable $K$-theory.

I have a couple of questions.

Why is 1. a good definition? I like my cohomology theories to be functorial under maps. Theory 1. clearly is not. I believe it is functorial with respect to proper maps. This reminds me of compactly supported cohomology. But why is this then not called compactly supported $K$-theory?

And:

How are 2. and 3. related?

More specifically:

What exactly does 3. describe? Are these virtual vector bundles that admit numerable trivializations?

finally:

Is there a reference where all these definitions are discussed?