Where could the proof of the following theorem be found : a manifold is metrizable if and only if it is paracompact
2026-02-22 21:26:10.1771795570
Proof a theorem about Metrizable manifold
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This is due to Smirnov (1951) originally (Russian paper), it follows from the following theorem (4.4.19 in Engelking, General Topology (2nd ed.)):
plus the "easy" direction that a metrisable space is paracompact.
It's also in my edition of Munkres (Topology, a first course; 1st edition) under the Smirnov Metrization Theorem (Theorem 5.1 on page 260). The newer edition has it page 261, Theorem 42.1. The quoted theorem 4.4.19, is exercise 23G in Willard, General Topology.