Hatcher-Algebraic-Topology · Problem 2.2.21

The Euler characteristic is fundamentally additive because it counts cells with alternating signs—a principle that breaks when regions overlap. The key insight is that when you sum $\chi(A) + \chi(B)$, you've double-counted every cell in $A \cap B$, so you must subtract $\chi(A \cap B)$ exactly once to correct for this overcounting. This mirrors the inclusion-exclusion principle: you're partitioning $X$'s cells into three disjoint groups (only in $A$, only in $B$, and in both) and verifying that the alternating sum respects this decomposition.

The Euler characteristic is fundamentally additive because it counts cells with alternating signs—a principle that breaks when regions overlap. The key insight is that when you sum $\chi(A) + \chi(B)$, you've double-counted every cell in $A \cap B$, so you must subtract $\chi(A \cap B)$ exactly once to correct for this overcounting. This mirrors the inclusion-exclusion principle: you're partitioning $X$'s cells into three disjoint groups (only in $A$, only in $B$, and in both) and verifying that the alternating sum respects this decomposition.