How does Hausdorff define compactness using infinite subsets and limit points?

Conditions

• E is a set.
• S is an arbitrary infinite subset of E (S ⊂ E).
• The limit point must belong to E.

Reasoning, step by step

1. Take any infinite subset S contained in E.
2. Identify the limit points of S.
3. Check if at least one limit point of S is an element of E.
4. If every infinite subset S has a limit point in E, then E is compact.

Example

The video displays the condition: 'A set E is compact if every infinite S ⊂ E has a limit pt in E,' accompanied by a diagram of a green subset S inside a larger gray set E.

Common misconceptions

• Thinking that the limit point must be an element of the subset S, rather than just the ambient set E.
• Assuming that finite subsets also need to have limit points in E.

Watch the explanation

Explore next

• What does it mean for a point to be a limit point in this context?
• Why is the closed interval [0,1] compact under the limit-point definition?