Minimal uncountable well-ordered set

This article describes a standard counterexample to some plausible but false implications. In other words, it lists a pathology that may be useful to keep in mind to avoid pitfalls in proofs
View other standard counterexamples in topology

Definition

The minimal uncountable well-ordered set is the well-ordered set corresponding to the minimal uncountable ordinal. This is viewed as a topological space by giving it the order topology.

Topological space properties

Properties it does satisfy

• Linearly orderable space
• Normal space
• Limit point-compact space: For full proof, refer: Minimal uncountable well-ordered set is limit point-compact

Properties it does not satisfy

• Compact space: For full proof, refer: Minimal uncountable well-ordered set is not compact