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Open Set, Closed Set: Definition

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An open set contains its boundary; it is a generalization of an open interval. A closed set does not contain its boundary and is a generalization of a closed interval. In topology, a closed set is defined an one whose complement is open.

Open Set / Closed Set Examples

Every open interval is also an open set [1]. For example, the interval (3, 5) is open because any x-value in the set will be between 3 and 5. In other words, if you choose a number very close to one of the boundaries (3 or 5), there will always be a set of numbers surrounding it that does not contain the boundary. Let’s say you choose 3.001. The numbers 3.00001 and 3.01 are:

  • In the interval [3, 5],
  • Surround 3.001,
  • Do not contain the boundary.

You could continue choosing number ad infinitum and never reach the boundary. This leads to an alternative definition of an open set, which is in terms of distance. A set (a, b) is open if it contains all numbers “sufficiently close” to a and b 2.

Properties of Open Set / Closed Set

  • The complement of an open set is closed. For example, [3, 5] is closed because its complement is two open sets
    (-∞ 3) ∪ (5, ∞).
  • Every union of open sets (the smallest set that contains both sets) is open.
  • Every finite intersection of open sets is open.

However, the fact that the complement of an open set is closed does not mean that “closed set” and “open set” are antonyms. Sets can be open, closed, both, or neither [3].

*Note: A complement is all elements, from a universal set, that are not in the set of interest. For example, if your universal set is {1, 2, 3, 4} then the complement of {1, 2} is {3, 4}.


References

[1] Knapp, A. (2005). Basic Real Analysis. Birkhäuser Boston.
[2] A Short Introduction to Metric Spaces: Section 1: Open and Closed Sets. Retrieved August 4, 2021 from: https://math.hws.edu/eck/metric-spaces/open-and-closed-sets.html
[3] Lamb, E. (2013). Can a Closed Set Be Open? Can an Open Set Be Closed? When Math and Language Collide. Retrieved August 4, 2021 from: https://blogs.scientificamerican.com/roots-of-unity/can-a-closed-set-be-open-can-an-open-set-be-closed-when-math-and-language-collide/

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Stephanie Glen. "Open Set, Closed Set: Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/open-set-closed-set-definition/
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