Sorgenfrey plane

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The Sorgenfrey plane is a well-known example in topology. It is built from two copies of the real line, each given the lower-limit topology (also called the Sorgenfrey line). In this setting, open sets on the plane are unions of rectangles [a, b) × [c, d), where [a, b) means all numbers from a up to but not including b. So the basic “buildings blocks” are half-open rectangles that extend to the right and upward.

Important properties:
- Each Sorgenfrey line is Lindelöf, but the Sorgenfrey plane is not Lindelöf. So a product of Lindelöf spaces need not be Lindelöf.
- The anti-diagonal Δ = { (x, -x) : x ∈ R } is an uncountable, discrete subset of the plane. This subset is non-separable, even though the plane itself is a separable space. This shows that separability does not always pass to closed subspaces.
- The sets K = { (x, -x) : x ∈ Q } and Δ \ K are closed in the plane, yet they cannot be separated by open sets. This demonstrates that the Sorgenfrey plane is not a normal space, i.e., the product of normal spaces need not be normal. In fact, even the finite product of perfectly normal spaces may fail to be normal.

In short, the Sorgenfrey plane is a key example that challenges several intuitive ideas about how products of nice spaces behave.