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How are open sets used to give a set a topological structure?

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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.

To give a set $X$ a topological structure, one must specify a collection of subsets $T$ , known as a topology, that satisfies three fundamental axioms:12

Existence: Both the set $X$ itself and the empty set must be members of $T$ .1
Finite Intersections: The intersection of any finite number of sets in $T$ must also be in $T$ .12
Arbitrary Unions: The union of any collection of sets in $T$ (even an infinite one) must be in $T$ .12

The subsets within this collection $T$ are defined as open sets. By establishing these rules, mathematicians can define concepts like continuity and limit points without relying on distance, allowing for the study of topological spaces in a highly generalized way.123