Alphabeta MathHow 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.
The one-point compactification of discrete is not
Statement refuted
The one-point compactification of discrete has the Stone–Čech extension property.
Facts & Assumptions
Given: Discrete , its one-point compactification , and for even and for odd .
A neighbourhood of in the one-point compactification is the complement of a closed compact subset of (The one-point (Alexandroff) compactification , whose open sets are the open sets of together with the complements in of the closed compact subsets of ).
In a discrete space, compact subsets are finite (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).
Counterexample
Suppose extends continuously to , and write .
The open interval contains , so continuity gives a neighbourhood of on which has values in that interval. By [L1] and [L2], this neighbourhood contains every natural except finitely many.
Both an even and an odd natural lie outside every finite subset of . Their -values are and , which cannot both belong to an interval of radius . This contradicts step 1.2.
Hence has no continuous extension; the asserted Stone–Čech property is false.
Depends on
- The one-point (Alexandroff) compactification $X^{*} = X \cup \{\infty\}$, whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$
- $X^{*}$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{*}$ exactly when $X$ is not compact; and $X^{*}$ is Hausdorff exactly when $X$ is locally compact and Hausdorff
- The Stone–Čech compactification by its compact-Hausdorff extension property
- The natural numbers $\mathbb{N}$ (von Neumann)
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
27 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
- E. Moorhouse, The Stone–Čech Compactification (standard reference, not scraped)