Difference between revisions of "Sorgenfrey plane"
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== Separation axioms == | == Separation axioms == | ||
| − | The Sorgenfrey plane is [[separation axioms|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 <math>A</math> of <math>-\Delta</math> is a closed subspace of <math>\mathbb{R}_l^2</math>, and it can be shown that there do not exist disjoint open sets about <math>A</math> and <math>-Delta \setminus A</math> in <math>\mathbb{R}_l^2</math>. | + | The Sorgenfrey plane is [[separation axioms|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 <math>A</math> of <math>-\Delta</math> is a closed subspace of <math>\mathbb{R}_l^2</math>, and it can be shown that there do not exist disjoint open sets about <math>A</math> and <math>-\Delta \setminus A</math> in <math>\mathbb{R}_l^2</math>. |
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[[Category:Topology]] | [[Category:Topology]] | ||
Latest revision as of 11:20, 26 May 2019
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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Countability axioms
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).
Separation axioms
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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