False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Refuted: every separable space is second countable

Statement

Every separable space is second countable.

Facts & Assumptions

Given: The lower-limit topology on $R$ , whose basic open sets are the intervals $[a, b)$ with $a < b$ .

[L1]

The rational numbers are at most countable and dense in the real line, and the real line is uncountable ( $Q$ is countably infinite, The rationals embed densely in the reals, $R$ is uncountable (Cantor's nested intervals, 1874)).

[L2]

A basis gives, for every open set $U$ and each $x \in U$ , a basis member $B$ with $x \in B \subseteq U$ (A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis).

[L3]

Separability means the existence of an at most countable dense subset, while second countability means the existence of an at most countable basis (Separability: the existence of an at most countable dense subset, Second countability: an at most countable basis for the topology).

[L4]

Every nonempty at most countable set can be enumerated by a surjection from $N$ (A nonempty set is at most countable iff it is a surjective image of $N$ ).

Refutation

technique · direct

1.1

Every nonempty basic interval $[a, b)$ meets $Q$ , so $Q$ is a countable dense subset of the lower-limit line.

L1L3

1.2

If an at most countable basis $B$ existed, it would be nonempty, so enumerate it as $(B_{n})$ by [L4]. For $x \in R$ , the set $E_{x} = {n : x \in B_{n} \subseteq [x, x + 1)}$ is nonempty by [L2]; let $n (x)$ be its least member. If $x < y$ and $n (x) = n (y)$ , then the common basis member contains $x$ but is contained in $[y, y + 1)$ , impossible. Thus $x \mapsto n (x)$ would inject $R$ into $N$ , contradicting [L1].

L1L2L4

2.1

Step 1.1 gives separability, whereas step 1.2 rules out an at most countable basis; thus this separable space is not second countable.

step 1.1step 1.2L3∎

Depends on

Used by

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

Dependency tree · two levels

49 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

UCR General Topology Notes (standard reference, not scraped)
Lower limit topology (Wikipedia) (standard reference, not scraped)