---
tag: Y4T3GVK8GV
kind: counterexample
name: Sorgenfrey line
msc: [54G20, 54A10]
uses: [0Z2P41SVDP, 0CP5NDD21C, 03PPNCM9GG, 091GJ3FYD7, SW0SN1NEJE, 0RJSBR9GW1]
sources:
  - url: https://en.wikipedia.org/wiki/Lower_limit_topology
---

## Counterexample

The Sorgenfrey line $\R_\ell$ is the set $\R$ with the topology generated by the base of half-open intervals $[a, b)$ with $a < b$.
It is [[separable|0Z2P41SVDP]] and [[first-countable|0CP5NDD21C]], but it is not [[second-countable|03PPNCM9GG]] and not [[metrizable|091GJ3FYD7]].

## Remarks

The rationals are dense, since each $[a, b)$ contains a rational number.
The intervals $[x, x + 1/n)$ for $n \ge 1$ form a countable neighbourhood base at $x$.
Let $\mathcal{B}$ be any base.
For each $x \in \R$ choose $B_x \in \mathcal{B}$ with $x \in B_x \subseteq [x, x + 1)$.
Then $x$ is the least element of $B_x$, so $x \mapsto B_x$ is injective and $\mathcal{B}$ is uncountable.
A separable metrizable space is second-countable, so $\R_\ell$ is not metrizable.
The Sorgenfrey line is finer than the usual topology, it is [[Lindelöf|SW0SN1NEJE]] and normal, and it is [[totally disconnected|0RJSBR9GW1]] because each $[a, b)$ is both open and closed.

## Explanations

### L1 human

Here an open set around a point may stretch to the right of the point but must start exactly at it.
This small change to the real line breaks the link between "countable dense set" and "countable base".
It is the standard example that separable spaces need not be metrizable even when they are very regular.