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:


1. 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.
2. 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.
3. 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?
> 
> 
> 



K-theory of non-compact spaces
2 and 3 are not equivalent, because of a phenomenon known as "phantom maps".You can have a map of a CW complex $X$ to $Y$ which is non-trivial in homotopy, but homotopy trivial when restricted to every finite subcomplex of $X$.


That this actually occurs for K-cohomology is shown in an old paper of Anderson and Hodgkin, "the K-theory of Eilenberg Maclane Complexes" 
 <http://www.sciencedirect.com/science/article/pii/0040938368900098>
See corollary 1 of that paper, which gives examples for Eilenberg-Maclane complexes.


Also, in answer to your question about an interpretation of 3 as virtual vector bundles, I don't think you will get any joy there. See the paper by Jackowski and Oliver quoted by Neil Strickland in his answer to this earlier MO question [Is there a good definition of (topological) K-theory over arbitrary spaces](https://mathoverflow.net/questions/73974/is-there-a-good-definition-of-topological-k-theory-over-arbitrary-spaces?rq=1)


