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How are closed sets related to open sets in topology?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In topology, the relationship between open and closed sets is fundamental: a set is defined as closed if and only if its complement is an open set.1

While a topological space is most commonly defined by a collection of open sets that satisfy specific axioms regarding unions and intersections, it can alternatively be defined using closed sets as the primary building blocks.21

The two concepts are intrinsically linked through the following properties:
* Complementarity: The complement of any open set is closed, and conversely, the complement of any closed set is open.1
* Axiomatic Duality: Just as the union of any collection of open sets is open and the intersection of a finite number of open sets is open, closed sets follow dual rules: the intersection of any collection of closed sets is closed, and the union of a finite number of closed sets is closed.
* Convergence: Closed sets are particularly significant in the study of limit points, as they contain all their limit points.21