◀ ▲ ▶Branches / Topology / Definition: Boundary Points, Closures, Interiors, and Exteriors

Definition: Boundary Points, Closures, Interiors, and Exteriors

Let $(X,\mathcal O)$ be a topological space and let $B\subset X$.

• A point $A\in X$ is called a boundary point of $B$ if every neighborhood $N_A$ has points in common with \(B\) and also with the set complement $B^C=X\setminus B.$

• The set of all boundary points of \(B\) is called the boundary of the set $B$ and denoted by \(\delta B\).
• The closure $B^-$ of $B$ is the set union of $B$ with its boundary $B\cup \delta B.$
• The interior $B^\circ$ of $B$ is the set difference of $B$ without its boundary $B\setminus \delta B.$
• The exterior $B^e$ of $B$ is the set complement of the closure of $B.$

Mentioned in:

Definitions: 1 2 3 4
Explanations: 5
Proofs: 6 7 8 9
Propositions: 10 11 12

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References

Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984