DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Smash product of based spaces

Definition

Let $X$ and $Y$ be based CGWH spaces. Their wedge inside the k-product is

$X \lor Y = (X \times {*_{Y}}) \cup ({*_{X}} \times Y) \subseteq X \times_{k} Y .$

The smash product is the based, kified quotient

$X \land Y = k ((X \times_{k} Y) / (X \lor Y)),$

based at the collapsed wedge. If $q : X \times_{k} Y \to X \land Y$ is the quotient map, then the quotient universal property says that a based map $X \land Y \to Z$ is equivalently a continuous map $X \times_{k} Y \to Z$ that is constant with value $*_{Z}$ on $X \lor Y$ . The kification does not change this test against a CGWH target.

We write $x \land y$ for $q (x, y)$ . No claim that $X \land Y$ is well-pointed is made without further hypotheses.

Depends on

Used by

Canonical associativity, symmetry, and unit maps for smash products Proposition

Dependency tree · two levels

15 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

J. P. May, A Concise Course in Algebraic Topology (standard reference, not scraped)