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In Hatcher's Algebraic Topology on page 140, he writes that it is easy to compute the cellular boundary map in the case $n=1$, since $$d_{1}:H_{1}(X^{1},X^{0}) \rightarrow H_{0}(X^{0})$$ is simply the same as the simplicial boundary map $$\Delta_{1}(X) \rightarrow \Delta_{0}(X).$$

Why exactly is this the case?

Added by TheGeekGreek: I would also like to address the question on the sense or interpretation of $\Delta_\ast(X)$, where $X$ is a cell complex, because not every cell complex is triangulable.

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