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 separable space is second countable

Statement

Every separable space is second countable.

Facts & Assumptions

Given: The lower-limit topology on $\mathbb R$ , 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 ( $\mathbb{Q}$ is countably infinite, The rationals embed densely in the reals, $\mathbb{R}$ 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 $\mathbb N$ (A nonempty set is at most countable iff it is a surjective image of $\mathbb{N}$ ).

Refutation

technique · direct

1.1

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

L1L3

1.2

If an at most countable basis $\mathcal B$ existed, it would be nonempty, so enumerate it as $(B_n)$ by [L4]. For $x\in\mathbb R$ , 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 $\mathbb R$ into $\mathbb N$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 98 results over 25 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)
Lower limit topology (Wikipedia) (standard reference, not scraped)