The Sorgenfrey plane is a useful example in topology. It is formed by taking the Cartesian product of the lower-limit topology on with itself.
Let be the anti-diagonal. It is a closed subspace of the Sorgenfrey plane, but it inherits the discrete topology as a subspace: consider the basis element given by .
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The Sorgenfrey plane is first-countable, is separable (has a countably dense subset, namely ), but not Lindelof, and consequently not second-countable (does not have a countable basis).
The Sorgenfrey plane is regular, but not normal. It is regular because it is the Cartesian product of regular spaces. It is not normal; we can see this because any subset of is a closed subspace of , and it can be shown that there do not exist disjoint open sets about and in .
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