Union of Closed Sets: Finite vs. Infinite Examples

Thread starter Bachelier
Start date Apr 5, 2011
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Closed Sets Union

In summary, a closed set is a set that contains all its limit points. This means that every convergent sequence of elements in the set will converge to an element within the set. The difference between a finite closed set and an infinite closed set is that a finite closed set has a limited number of elements, while an infinite closed set has an unlimited number of elements. An infinite closed set can contain a finite closed set, as a finite set is still considered closed. Examples of finite closed sets include a single point, a line segment, a triangle, a rectangle, and a circle. Examples of infinite closed sets include the set of all real numbers, the set of all integers, and the set of all rational numbers. Other examples include a

Apr 5, 2011

Bachelier

A Union of a FINITE collection of closed sets is closed. But if it is an infinite collection?
Can someone provide an example please?

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Apr 5, 2011

micromass

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The standard counterexample is this: [tex]\bigcup_{n\geq 1}{[1/n,1]}[/tex]. This is a union of closed sets, but the union equals ]0,1] and this is not closed.

Apr 5, 2011

Bachelier

Good. Thanks.

1. What is the definition of a closed set?

A closed set is a set that contains all its limit points, which means that every convergent sequence of elements in the set converges to an element within the set.

2. What is the difference between a finite closed set and an infinite closed set?

A finite closed set contains a limited number of elements, while an infinite closed set contains an unlimited number of elements. In other words, a finite closed set has a specific, defined size, while an infinite closed set does not.

3. Can an infinite closed set contain a finite closed set?

Yes, an infinite closed set can contain a finite closed set. This is because a finite set is still considered closed, as it contains all its limit points, even if it is contained within a larger, infinite closed set.

4. What are some examples of finite closed sets?

Examples of finite closed sets include a single point, a line segment, a triangle, a rectangle, and a circle.

5. What are some examples of infinite closed sets?

Examples of infinite closed sets include the set of all real numbers, the set of all integers, and the set of all rational numbers. Other examples include a closed interval on the real number line, a closed ball in n-dimensional Euclidean space, and a closed half-plane in the Cartesian plane.