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Under choice, the lower-limit line is regular and separable but not second countable and therefore not metrizable
Example
Assume the Axiom of Choice. The lower-limit line is regular and separable, but not second countable and hence not metrizable.
Facts & Assumptions
Given: The lower-limit topology on and the Axiom of Choice.
The lower-limit line is regular, and its basic intervals are (The lower-limit line has a clopen basis, is regular, and is Lindelöf under countable choice, The lower-limit topology on , with the half-open intervals as a basis).
The rationals are countable and meet every nonempty usual interval, hence every ( is countably infinite, The rationals embed densely in the reals).
Under choice, a metrizable space is second countable exactly when it is separable (Assuming countable choice, a metrizable space is second countable if and only if it is separable if and only if it is Lindelöf).
Verification
By [L1] the space is regular, and by [L2] the countable set is dense, so it is separable.
Suppose is a basis. For each real , the basis condition for yields a least-index with .
If and , then , impossible. Thus injects into , contradicting is uncountable (Cantor's nested intervals, 1874).
Therefore the lower-limit line is not second countable. If it were metrizable, its separability from step 1.1 and [L3] would make it second countable, another contradiction.
Depends on
- The lower-limit topology on $\mathbb{R}$, with the half-open intervals $[a,b)$ as a basis
- The lower-limit line has a clopen basis, is regular, and is Lindelöf under countable choice
- Separability: the existence of an at most countable dense subset
- Second countability: an at most countable basis for the topology
- Assuming countable choice, a metrizable space is second countable if and only if it is separable if and only if it is Lindelöf
- $\mathbb{Q}$ is countably infinite
- The rationals embed densely in the reals
- $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)
- The Axiom of Choice
Used by
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Sources
- L. A. Steen and J. Seebach, Counterexamples in Topology (standard reference, not scraped)