DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Skeletal mapping telescope of a cw pair

Definition

For a CW pair $(X, A)$ let $X^{i}$ be its $i$ -skeleton and $A^{i} = A \cap X^{i}$ , following Skeleta, CW subcomplexes, and relative CW complexes. Its skeletal mapping telescope is the CW pair $T_{X} = ⋃_{i \geq 0} X^{i} \times [i, \infty), T_{A} = ⋃_{i \geq 0} A^{i} \times [i, \infty) .$ Give $[0, \infty)$ vertices at the nonnegative integers and use the CW weak topology on these subcomplexes of $X \times [0, \infty)$ . Projection $p (x, t) = x$ defines a continuous map $(T_{X}, T_{A}) \to (X, A)$ . The telescope of the empty space is empty.

Depends on

Skeleta, CW subcomplexes, and relative CW complexes

Used by

The skeletal telescope projects by a homotopy equivalence of pairs Lemma

Dependency tree · two levels

2 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

Hatcher, Algebraic Topology, Lemma 2.34, telescope construction p.138 (standard reference, not scraped)
May, A Concise Course in Algebraic Topology, 14§6, telescope construction pp.114–116 (standard reference, not scraped)