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In Prasolov's book "Elements of Homology Theory" the author want prove the following theorem.

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In order to do this, he needs to show that the theorem it's true for any open set $U\subseteq \mathbb{R}$. There are two steps that in my opinio contain some mistakes.

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  1. I don't understand how it's possible that $K_i\subseteq K_{i+1}$.
  2. By construction we have not $K_i\setminus K_{i-1}\subseteq L_i\setminus L_{i-1}$.

Are these mistakes or I'm doing it wrong?

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    I think that the author made a mistake in the definition of $K_2$. What he really wanted to do is "since $K_1$ is a compact subset of $U$ then it can be covered by finitely many sets $V_{i_1},\ldots, V_{i_m}$". So covering entire $K_1$ makes more sense then covering its boundary only. And with that $K_i\subset\mbox{int} K_{i+1}$ actually holds (in addition to all other properties). – 2017-02-26

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