Some authors define a manifold as a paracompact Hausdorff space that is locally Euclidean. Also it is said that a product of two manifolds is a manifold. However, we know that product of a two paracompact spaces is not necessarily paracompact. So how can we be sure that a product of two manifolds is also paracompact and thus is also a manifold? Is it somehow related to the second-countability property that is usually defined along paracompactness?
2026-02-23 02:40:11.1771814411
Why the product of two manifolds is paracompact?
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By the Smirnov metrization theorem, a paracompact Hausdorff space that is locally metrizable is metrizable. Therefore every manifold is metrizable, and hence so is the product of two manifolds. In particular the product is paracompact.