Unformatted text preview: ) ⊂ B , then A open = ⇒ A · B open . Let a ∈ A and b ∈ B . If b 6 = 0, and ( a±, a + ± ) ⊂ A for some ± > 0, then ( ab  b  ±, ab +  b  ± ) ⊂ A · B. Now suppose b = 0 and (±, ± ) ⊂ B for some ± > 0. If a ∈ A satisﬁes a 6 = 0, then ( a  ±,  a  ± ) ⊂ A · B , providing a neighborhood of ab = 0 contained in A · B in that case. If a = 0, then there exists δ > 0 with (δ, δ ) ⊂ A , and then (±δ, ±δ ) ⊂ A · B . 4.1.5.2 The set A is closed because A = A 1 ∩ A 2 ∩ ··· ∩ A n , where A i is the set deﬁned by f i ( x ) = 0, and each A i is the inverse image of the closed set { } by a continuous map, hence closed. It need not be compact, of course: consider the case n = 1 and f ( x ) = sin πx , so A = Z . 1...
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 Fall '07
 HUBBARD
 Topology, Sets, Empty set, Open set, Topological space, iy, Ix Iy

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