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.
Refuted: Lindelöfness is hereditary
Statement
Lindelöfness is hereditary.
Facts & Assumptions
Given: The uncountable discrete space and its one-point compactification .
The one-point compactification is compact and contains as an open subspace ( is compact and contains as an open subspace; is dense in exactly when is not compact; and is Hausdorff exactly when is locally compact and Hausdorff).
Compactness gives a finite subcover for every open cover, Lindelöfness gives an at most countable subcover, and a property is hereditary when every subspace has it (Countably compact, Lindel"of, sequentially compact, limit point compact and -compact spaces, and relatively compact subsets, Hereditary, open-hereditary and closed-hereditary properties of topological spaces).
The real line is uncountable and every subset of a discrete space is open ( is uncountable (Cantor's nested intervals, 1874), The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).
A space is locally compact when every point has a compact neighbourhood, and Hausdorff when distinct points have disjoint open neighbourhoods (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not).
Refutation
The discrete space is Hausdorff because distinct singleton neighbourhoods are disjoint, and locally compact because each point has the compact singleton neighbourhood; it is not compact because its singleton cover has no finite subcover. Thus its one-point compactification has the usual compact Hausdorff behavior, and in any case [L1] makes compact with as an open subspace. By [L2], is Lindelöf.
The subspace is discrete and has the open cover ; any subcover must contain every singleton, so no at most countable subfamily covers the uncountable set .
Thus the Lindelöf space has the non-Lindelöf subspace , so Lindelöfness is not hereditary.
Depends on
- Countably compact, Lindel\"of, sequentially compact, limit point compact and $\sigma$-compact spaces, and relatively compact subsets
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- 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 discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
- $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)
- Hereditary, open-hereditary and closed-hereditary properties of topological spaces
Used by
Dependency tree · two levels
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Sources
- UCR General Topology Notes (standard reference, not scraped)
- Fort space (Wikipedia) (standard reference, not scraped)
- Alexandroff extension (Wikipedia) (standard reference, not scraped)