Lemma 5.13.5. Let $X$ be a Hausdorff and quasi-compact space. Let $X = \bigcup _{i \in I} U_ i$ be an open covering. Suppose given an integer $p \geq 0$ and for every $(p + 1)$-tuple $i_0, \ldots , i_ p$ of $I$ an open covering $U_{i_0} \cap \ldots \cap U_{i_ p} = \bigcup W_{i_0 \ldots i_ p, k}$. Then there exists an open covering $X = \bigcup _{j \in J} V_ j$ and a map $\alpha : J \to I$ such that $\overline{V_ j} \subset U_{\alpha (j)}$ and such that each $V_{j_0} \cap \ldots \cap V_{j_ p}$ is contained in $W_{\alpha (j_0) \ldots \alpha (j_ p), k}$ for some $k$.

Proof. Since $X$ is quasi-compact, there is a reduction to the case where $I$ is finite (details omitted). We prove the result for $I$ finite by induction on $p$. The base case $p = 0$ is immediate by taking a covering as in Lemma 5.13.4 refining the open covering $X = \bigcup W_{i_0, k}$.

Induction step. Assume the lemma proven for $p - 1$. For all $p$-tuples $i'_0, \ldots , i'_{p - 1}$ of $I$ let $U_{i'_0} \cap \ldots \cap U_{i'_{p - 1}} = \bigcup W_{i'_0 \ldots i'_{p - 1}, k}$ be a common refinement of the coverings $U_{i_0} \cap \ldots \cap U_{i_ p} = \bigcup W_{i_0 \ldots i_ p, k}$ for those $(p + 1)$-tuples such that $\{ i'_0, \ldots , i'_{p - 1}\} = \{ i_0, \ldots , i_ p\}$ (equality of sets). (There are finitely many of these as $I$ is finite.) By induction there exists a solution for these opens, say $X = \bigcup V_ j$ and $\alpha : J \to I$. At this point the covering $X = \bigcup _{j \in J} V_ j$ and $\alpha$ satisfy $\overline{V_ j} \subset U_{\alpha (j)}$ and each $V_{j_0} \cap \ldots \cap V_{j_ p}$ is contained in $W_{\alpha (j_0) \ldots \alpha (j_ p), k}$ for some $k$ if there is a repetition in $\alpha (j_0), \ldots , \alpha (j_ p)$. Of course, we may and do assume that $J$ is finite.

Fix $i_0, \ldots , i_ p \in I$ pairwise distinct. Consider $(p + 1)$-tuples $j_0, \ldots , j_ p \in J$ with $i_0 = \alpha (j_0), \ldots , i_ p = \alpha (j_ p)$ such that $V_{j_0} \cap \ldots \cap V_{j_ p}$ is not contained in $W_{\alpha (j_0) \ldots \alpha (j_ p), k}$ for any $k$. Let $N$ be the number of such $(p + 1)$-tuples. We will show how to decrease $N$. Since

$\overline{V_{j_0}} \cap \ldots \cap \overline{V_{j_ p}} \subset U_{i_0} \cap \ldots \cap U_{i_ p} = \bigcup W_{i_0 \ldots i_ p, k}$

we find a finite set $K = \{ k_1, \ldots , k_ t\}$ such that the LHS is contained in $\bigcup _{k \in K} W_{i_0 \ldots i_ p, k}$. Then we consider the open covering

$V_{j_0} = (V_{j_0} \setminus (\overline{V_{j_1}} \cap \ldots \cap \overline{V_{j_ p}})) \cup (\bigcup \nolimits _{k \in K} V_{j_0} \cap W_{i_0 \ldots i_ p, k})$

The first open on the RHS intersects $V_{j_1} \cap \ldots \cap V_{j_ p}$ in the empty set and the other opens $V_{j_0, k}$ of the RHS satisfy $V_{j_0, k} \cap V_{j_1} \ldots \cap V_{j_ p} \subset W_{\alpha (j_0) \ldots \alpha (j_ p), k}$. Set $J' = J \amalg K$. For $j \in J$ set $V'_ j = V_ j$ if $j \not= j_0$ and set $V'_{j_0} = V_{j_0} \setminus (\overline{V_{j_1}} \cap \ldots \cap \overline{V_{j_ p}})$. For $k \in K$ set $V'_ k = V_{j_0, k}$. Finally, the map $\alpha ' : J' \to I$ is given by $\alpha$ on $J$ and maps every element of $K$ to $i_0$. A simple check shows that $N$ has decreased by one under this replacement. Repeating this procedure $N$ times we arrive at the situation where $N = 0$.

To finish the proof we argue by induction on the number $M$ of $(p + 1)$-tuples $i_0, \ldots , i_ p \in I$ with pairwise distinct entries for which there exists a $(p + 1)$-tuple $j_0, \ldots , j_ p \in J$ with $i_0 = \alpha (j_0), \ldots , i_ p = \alpha (j_ p)$ such that $V_{j_0} \cap \ldots \cap V_{j_ p}$ is not contained in $W_{\alpha (j_0) \ldots \alpha (j_ p), k}$ for any $k$. To do this, we claim that the operation performed in the previous paragraph does not increase $M$. This follows formally from the fact that the map $\alpha ' : J' \to I$ factors through a map $\beta : J' \to J$ such that $V'_{j'} \subset V_{\beta (j')}$. $\square$

## Comments (4)

Comment #652 by Wei Xu on

Typos,

should be

and

"set $V'_{j_0} = V_{j_0} \setminus \overline{V_{j_1}} \cup \ldots \cup \overline{V_{j_p}}$."

should be

"set $V'_{j_0} = V_{j_0} \setminus (\overline{V_{j_1}} \cap \ldots \cap \overline{V_{j_p}})$."

Comment #669 by We on

Thanks. But "cup" should be "cap", not just only adding $()$.

Comment #670 by on

OK, I should have realized this because I was wondering why you just wanted to add some brackets... Anyway, I tried again, see here. Thanks!

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09UW. Beware of the difference between the letter 'O' and the digit '0'.