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On P.38 of Max Karoubi's 'K-Theory: An Introduction', he attempts to prove that for a compact space $X$,

$[X, BGL_n(k)] \cong \Phi^k_n(X)$

where $Proj_n(k^N)=\{p \in GL_N(k):p^2=p, p$ has rank $n\}$, $BGL_n(k)=\varinjlim Proj_n(k^N)$, $[X, Y]=$set of homotopy classes of continuous maps from $X$ to $Y$, $\Phi^k_n(X)$ is the set of isomorphic classes of vector bundles of dimension $n$ over $X$

He already has proved the 'difficult' part that $\varinjlim [X,Proj_n(k^N)]\cong \Phi^k_n(X)$, thus it remains to show $\varinjlim [X,Proj_n(k^N)]= [X,\varinjlim Proj_n(k^N)]$. To justify this step, Karoubi writes "since $X$ is compact and $Proj_n(k^N)$ is closed in $Proj_n(k^{N^{'}})$ for $N \leq N'$". I do not follow how he can do this.

I am fairly certain that the statement "if $X$ is compact and $C_1 \subset C_2 \subset...$, with each $C_k$ closed in $\varinjlim C_n=\cup_{n=1}^{\infty} C_n$, then $\varinjlim [X,C_n]=[X,\cup_{n=1}^{\infty} C_n]$" is false; what is required for the conclusion to hold is that every compact set be contained in some $C_n$, which is true if each $C_k$ is OPEN in $\cup_{n=1}^{\infty} C_n$. I also realize that if each $C_N=Proj_n(k^N)$ has an open neighborhood that can be deformation retracted to it, then the conclusion holds, but I cannot see if this is the case here.

I would like some help regarding how to justify this step of passing the direct limit inside $[X,\cdot]$. Thanks in advance.

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    $\begingroup$ Hi, I am not sure if you already figured this out. Actually, I have the same question. So you might want to take a look at math.stackexchange.com/questions/2394787/… if you still get stuck on this problem. $\endgroup$ – YYF Aug 16 '17 at 14:51

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