CW complex

An $𝑛$ -dimensional CW complex $𝑋 𝑛$ is a topological space formed by gluing $𝑛$ -cells ( $𝑛$ -discs, up to homeomorphism) at their boundaries to a “skeleton” $(𝑛 - 1)$ -dimensional CW complex $𝑋 𝑛 - 1$ .

\emptyset = 𝑋 - 1 ↪ 𝑋 0 ↪ 𝑋 1 ↪ \dots

More formally, given an indexed family of continuous maps $𝜑 𝛼 : 𝕊 𝑛 - 1 \to 𝑋 𝑛 - 1$ , we form $𝑋 𝑛$ as the fibre coproduct

i.e. the quotient space

𝑋 𝑛 = 𝑋 𝑛 - 1 ⨿ ∐ 𝛼 𝔻 𝑛 \sim

by the relation induced by $𝜄 𝛼 (𝑥) \sim 𝜑 𝛼 (𝑥)$ for all $𝑥 \in 𝕊 𝑛 - 1 = 𝜕 𝔻 𝑛$ . A more general (i.e. possibly infinite dimensional) CW complex is the colimit of a sequence of CW complexes. #m/def/topology

#state/tidy | #lang/en | #SemBr