Suppose X is a topological space whose topology is coherent with a family B of subspaces.  Prove that if Y is another topological space, then a map f : X → Y is continuous if and only if f|B is continuous for every B ∈ B, where f|B is the restriction of f to B.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.1: Vector Spaces And Subspaces
Problem 44EQ
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Suppose X is a topological space whose topology is coherent with a family B of subspaces. 

Prove that if Y is another topological space, then a map f : X → Y is continuous if and only if f|B is continuous for every B ∈ B, where f|B is the restriction of f to B. 

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