This is a question about a proof in the article *Group Cohomology with Lipschitz Control and Higher Signatures* by Connes, Gromov and Moscovici, which may be found here. Namely, their "metric extension lemma" asserts that one can extend a metric from a subspace of a product to the whole space.

The precise statement is as follows and may be found on page 41 of the article:

Let $Y$ and $P$ be locally compact metrizable spaces, and let $Z\subset Y\times P$ be closed. Let $d$ be a metric on $Z$ such that $d|Y\times p=d_Y$ for all $p\in P$ and a given metric $d_Y$ on $Y=Y\times p$. Then there exists a metric $\bar d$ on $Y\times P$ such that (i) $\bar d|Y\times p=d_Y$ for all $p\in P$ and (ii) $\bar d|Z\geq d$.

I added the condition that $P$ is metrizable and that $Z$ is closed to the statement of the theorem, and both conditions are obviously necessary (Otherwise, one could, for instance, take $Z=(0,1)\subset[0,1]=P$ and $Y=\{*\}$, and the lemma would imply that every metric on $(0,1)$ can be bounded above by a metric on $[0,1]$.)

In the article, the authors first prove the statement in the case that $Y$ and $P$ are compact. This proof seems to be OK to me.

However, when they want to reduce the general case to the compact one, they take a locally finite cover of $Y\times P$ by compact product subsets $Y_i\times P_i$ (as YCor pointed out, every metric space is paracompact.) and equip each of them with a metric as constructed in the compact case. Then they define the final metric as the supremum of the metrics which are bounded above by the constructed metric on every $Y_i\times P_i$, and which equal $d_Y$ on every slice $Y\times p$.

I have three questions regarding this proof:

- I think one could have arbitrary metrics on $Y_i\times P_i$ if this product is small enough to contain only one point of $Z$. So how can the result have anything to do with $Z$?
- Can one explicitly describe this "supremum metric"? To be precise, how does one know that the set of metrics with the given properties is non-empty?
- Can we still prove the lemma in the general case, perhaps adding some condition on $Z$, say, $Z\subset Y\times p$ being coarsely dense for every $p\in P$?