False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Refuted: every first countable space is second countable

Statement

Every first countable space is second countable.

Facts & Assumptions

Given: The set $D=\mathbb R$ carrying the discrete topology.

[L1]

Every subset of a discrete space is open (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

[L2]

A space is first countable when every point has an at most countable local base, and second countable when it has an at most countable global basis (First countable space: a countable neighbourhood base at every point, Second countability: an at most countable basis for the topology).

[L3]

The real line is uncountable ( $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)).

Refutation

technique · direct

1.1

For each $x\in D$ , the one-member family $\{\{x\}\}$ is a local base, because every neighbourhood of $x$ contains the open singleton $\{x\}$ by [L1].

L1L2

1.2

If $\mathcal B$ is any basis of $D$ , then for each $x\in D$ some $B_x\in\mathcal B$ satisfies $x\in B_x\subseteq\{x\}$ , so $B_x=\{x\}$ and $\mathcal B$ contains every singleton.

2.1

Step 1.1 makes $D$ first countable, while steps 1.2 and [L3] make every basis uncountable; hence $D$ is not second countable.

step 1.1step 1.2L2L3∎

Depends on

Used by

Implication, preservation, counterexample, and choice ledger for the countability axioms Remark

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 61 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

UCR General Topology Notes (standard reference, not scraped)
First-countable space (Wikipedia) (standard reference, not scraped)
Second-countable space (Wikipedia) (standard reference, not scraped)