Boundary (topology) - Wikipedia
https://en.wikipedia.org/wiki/Boundary_(topology)A boundary point of a set refers to any element of that set's boundary. The boundary defined above is sometimes called the set's topological boundary to distinguish it from other similarly named notions such as the boundary of a manifold with boundary or the boundary of a manifold with corners, to name just a few example. Properties
Section 17. Closed Sets and Limit Points
faculty.etsu.edu › gardnerr › 5357Jun 02, 2016 · Definition. A set A intersects a set B if A ∩ B 6= ∅. An open set U ⊂ X is a neighborhood of x ∈ X if x ∈ U. Theorem 17.5 Let A be a subset of the topological space X. (a) Then x ∈ A if and only if every neighborhood of x intersects A. (b) Supposing the topology of X is given a basis , then x ∈ A if and only if every
Singleton (mathematics) - Wikipedia
https://en.wikipedia.org/wiki/Singleton_(mathematics)Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and {1} are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as {{1, 2, 3}} is a singleton as it contains a single element (which itself is a set, however, not a singleton).