I am currently reading up on K-theory and I am a bit confused by the definition of the topological K-Theory-spectrum for C*-algebras. It is mentioned in many articles as basic, but I can only find one definition which I am confused by. It goes like this: Let A be a $C^*$-algebra. The topoligcal K-spectrum $K^{top}(A)$ is the spectrum where every second space is $$GL(A)=\text{colim}GL_n(A).$$ The spectrum is then given by homotopy-equivalences $$K_0(A) \times BGL(A)\rightarrow \Omega GL(A)$$ and $$GL(A) \rightarrow \Omega (BGL(A))=\Omega (K_0(A) \times BGL(A))$$ that arise from the Bott periodicity theorem.
I want to understand how this agrees with the definition of a spectrum as can be found on Wikipedia. So if every second space is GL(A), what are the other spaces. What are the base points and the structure maps? Thank you