I am reading the first chapter of the book "Complex topological K-theory" by Efton Park, which is in general very good. However, for some reasons, which I don't understand, when working with compact Hausdorff spaces, he assumes that the space has finitely many connected components. This is an implicit assumption e.g. in the proof of Proposition 1.5.19, or explicit in the proof of Proposition 1.7.9.
Is there some reason for this restriction, or is the book seriously flawed?
EDIT: Let me state as an example the first of these propositions: Let $A$ be a closed subspace of a compact Hausdorff space $X$, let $V$ and $W$ be vector bundles over $X$, and suppose $\sigma: V\upharpoonright A\rightarrow W\upharpoonright A$ is a bundle homomorphism. Then $\sigma$ can be extended to a bundle homomorphism $\tilde{\sigma}: V\rightarrow W$.
The proof starts with the sentence: Without loss of generality, we assume that $X$ is connected; otherwise, work with $X$ one component at a time.
This is fine if $X$ has finitely many components. Not so fine if $X$ is e.g. the Cantor set and each component is a singleton.
EDIT2: The propositions mentioned above are (as indicated by the "e.g.") just examples. They are not the only affected results in the book. The same "problem" repeats throughout Sections 1.5 and 1.7 and I cannot state all the propositions here. So the question is really targeted to those who have and have read the book.