# Why the product of two manifolds is paracompact?

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?