I already know the Pasting Lemma:
Theorem (Pasting Lemma). Let $X,Y$ be topological spaces, and let $\{U_\alpha \}$ be either an arbitrary open cover of $X$, or a locally finite closed cover of $X$. Suppose that for each $\alpha$, there is a continuous map $f_\alpha : U_\alpha \to Y$ such that $f_\alpha = f_\beta$ on $U_\alpha \cap U_\beta$ whenever $U_\alpha \cap U_\beta$ is nonempty. Then the $f_\alpha $'s extend uniquely to a continuous map $f:X \to Y$.
On the other hand, I want to construct a continuous map whose domain is a 1-dimensional CW complex, i.e., a graph $X$. I already constructed a continuous map on each of the closed edges of $X$, but I cannot paste them using the pasting lemma, because maybe there would be a vertex that meets infinitely many edges. Is there any another method can I use?
Best Answer
The Pasting Lemma is inadequate in this case. However, a CW-complex $X$ has the "weak topology" with respect to its closed cells which means a subset $U \subset X$ is open in $X$ iff $U \cap e$ is open in $e$ for all closed cells $e$.
As a conseqeunce a function $f : X \to Y$ is continuous iff $f \mid_e : e \to Y$ is continuous for all closed cells $e$. You can apply this to your graph.