Why does removing the endpoint 0 make the half-open interval (0,1] noncompact?

Conditions

• E is the half-open interval (0,1].
• S is the infinite subset {$1/n$ : $n=1$,2,3,...}.
• Compactness is judged by the limit-point definition.

Reasoning, step by step

1. Take the set E = (0,1].
2. Consider the infinite subset $S = {1/n}$ which is contained in E.
3. Identify the limit point of S, which is 0.
4. Check if the limit point 0 belongs to E.
5. Since 0 ∉ (0,1], the condition fails, proving E is not compact.

Example

The video changes the interval from [0,1] to (0,1], keeping the same subset {$1/n$}, and shows the label changing to 'not compact' because the limit point 0 is now outside the set.

Common misconceptions

• Assuming that compactness only depends on the set being bounded, ignoring the necessity of containing its limit points.
• Thinking that the subset {$1/n$} changes when the ambient set changes.

Watch the explanation

Explore next

• What is the difference between Fréchet compactness and limit-point compactness?
• How is compactness defined in a general topological space?