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Dec 9 at 16:05 comment added Matthew Kvalheim I believe so. Thank you, +1, accepted. However, now I realize that I did not ask the question I wanted to ask. I was trying to simplify Ferry's use of open covers in defining "small fibers" by using a Riemannian metric, which inadvertently permits your example due to noncompactness. I am actually only interested in examples for which, if $M$ is noncompact, $M$ is the interior of a compact manifold with boundary $L$ and the metric on $M$ is the restriction of one on $L$. I suppose it may be best for me to post this as a new question. Thank you for answering my misguided question and helping me.
Dec 9 at 16:00 vote accept Matthew Kvalheim
Dec 9 at 0:22 history answered alesia CC BY-SA 4.0