False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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An arbitrary disjoint union of second-countable manifolds need not be second-countable

Statement

False claim: an arbitrary disjoint union of second-countable manifolds is second-countable.

Facts & Assumptions

Given: An uncountable set $I$ and the disjoint union $X = ⨆_{i \in I} {*_{i}}$ of one-point manifolds.

[F1]

In the disjoint union topology, a subset of $X$ is open exactly when each trace on each summand is open (The disjoint union (coproduct) $⨆_{i} X_{i}$ with the final topology of the canonical injections: a set is open exactly when each of its traces is).

[F2]

A space is second countable when it has an at most countable basis (Second countability: an at most countable basis for the topology).

[L1]

The countable-union theorem on the A page requires the index set to be at most countable (Countable disjoint unions of fixed-dimensional smooth manifolds are smooth manifolds).

Refutation

technique · direct

1.1

Each singleton ${κ_{i} (*_{i})}$ is open in $X$ : its trace on the [F1] $i$ -th summand is the whole one-point space, and on every other summand it is empty, so [F1] makes it open. Thus $X$ is an uncountable discrete space.

2.1

If $B$ were a basis of $X$ , then for each $i \in I$ the open set [step 1.1, assume-hyp] ${κ_{i} (*_{i})}$ would contain some $B_{i} \in B$ with $κ_{i} (*_{i}) \in B_{i} \subseteq {κ_{i} (*_{i})}$ , forcing $B_{i} = {κ_{i} (*_{i})}$ . Distinct points therefore require distinct basis elements, so every basis is uncountable.

step 1.1assume-hyp

3.1

Hence $X$ is not second countable by [F2]. This is exactly why [L1] keeps the countability hypothesis explicit.

F2L1step 2.1∎

Depends on

Used by

An uncountable disjoint union of points is not second-countable Counterexample

Dependency tree · two levels

16 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

Rob van der Vorst, Introduction to differentiable manifolds, §1 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)