PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-07

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Euler characteristic is additive for finite CW pairs

Statement

For a finite CW pair $(X, A)$ , $χ (X) = χ (A) + \sum_{n} (- 1)^{n} # {n -cells of X ∖ A}$ . If $A \neq \emptyset$ , then $χ (X / A) = χ (X) - χ (A) + 1$ .

Facts & Assumptions

Given: A finite CW subcomplex $A \subseteq X$ .

Proof

technique · direct

1.1given

The cells of $X$ are the disjoint union of the cells of $A$ and those outside $A$ , so the first equality follows by splitting the finite alternating sum.

2.1step 1.1∎

If $A$ is nonempty, $X / A$ has one new $0$ -cell (the quotient point) and exactly the cells outside $A$ otherwise; its alternating count gives the second formula.

Depends on

Used by

Finiteness and additivity of the Euler characteristic Proposition

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Allen Hatcher, Algebraic Topology, Section 2.2 (standard reference, not scraped)