Pasting Lemma (Elaboration of Wikipedia’s proof)

The proof of the Pasting Lemma at Wikipedia is correct, but a bit unclear. In particular, it does not clearly show how the hypothesis that X, Y are both closed is being used. It actually has something to do with subspace topology.

I have added some clarifications here:

Pasting Lemma (Statement)

Let X, Y be both closed (or both open) subsets of a topological space A such that A=X\cup Y, and let B also be a topological space. If both f|_X: X\to B and f|_Y: Y\to B are continuous, then f:A \to B is continuous.

Proof:

Let U be a closed subset of B. Then f^{-1}(U)\cap X is closed in X since it is the preimage of U under the function f|_X:X\to B, which is continuous. Hence f^{-1}(U)\cap X=F\cap X for some set F closed in A. Since X is closed in A, f^{-1}(U)\cap X is closed in A.

Similarly, f^{-1}(U)\cap Y is closed (in A). Then, their union f^{-1}(U) is also closed (in A), being a finite union of closed sets.

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One Response to Pasting Lemma (Elaboration of Wikipedia’s proof)

  1. LispMobile says:

    You can contribute by editing that Wikipedia proof with your improvement.

    Like

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