Gluing Lemma

Definition

Consider topological Spaces $(X,\mathcal{T}),(Y,{\mathcal{T}}_Y)$, subsets $A,B\subset X\text{s.t.}(X\setminus A),(X\setminus B)\in \mathcal{T}\text{and}X=A\cup B$, and functions $(f:A\to Y),(g:B\to Y)$. Then, $f,g\in C^0\text{and}(\forall x\in A\cap B,f(x)=g(x))⟹\left(h:X\to Y,h(x)=\{\begin{array}{ll}f(x)& x\in A\\ g(x)& x\in B\end{array}\right)\in C^0$ Two continuous functions whose domains are closed, and with overlapping areas having the same values can be glued together to form a Continuous Function.