Let \(X\) and \(Y\) be topological spaces and \(\{A_i\}\) an arbitrary finite open cover of \(X\) or a finite closed cover of \(X\). Suppose that we are given continuous functions \(f_i\colon A_i\to Y\) that agree on overlaps. Then there exists a unique continuous function \(f\colon X\to Y\) whose restriction to each \(A_i\) equals \(f_i\). [1, Lemma 3.23]

References Link to heading

1. J. Lee, Introduction to Topological Manifolds. New York, NY: Springer New York, 2011. doi:10.1007/978-1-4419-7940-7