Boundary (topology)

In the mathematical branch of topology, the term edge is an abstraction of the intuitive notion of a limit of a range.

Definition

By definition, the boundary of a subset of a topological space the set difference between financial statements and interior of. The boundary of a set is commonly referred to, thus:

The points made ​​are called boundary points.

Explanation

Each boundary point of is also the point of contact of and every touch point is an element of or boundary point of. The points of contact together to form the concluding. It is therefore

For each subset of the topological space decomposes into the inside of the rim and from the appearance of:

Demarcation

But that related divergent boundary terms are there in algebraic topology and the theory of bounded manifolds.

Properties

  • The boundary of a set is always complete.
  • The boundary of a set consisting of exactly the points for which it holds that each of its points from both environments as well as points that do not lie in contains.
  • The boundary of a set is always equal to the edge of its complement.
  • The boundary of a set is the intersection of the conclusion of the crowd with the completion of their target.
  • A lot is just complete when it contains its boundary.
  • A lot is exactly then open if it is disjoint from its edge.
  • A lot is exactly then open and closed when its edge is empty.
  • Let be a topological space, an open subset with the subspace topology and a subset. Then the edge of in the same section of the edge from to. 's Leaving aside the requirement of openness of fall, the corresponding statement itself then does not apply if a subset of is, as the example shows.

Examples

  • Is an open or closed disk in the plane, as is the edge of the corresponding circle.
  • The edge of a subset of is all about.

Edge axioms

For a topological space forming the rim is a set operator, the power set of. This met and remember to follow the four rules, the so-called edge axioms:

Through the four rules (R1 ) - ( R4) the structure of the topological space is clearly defined. The given by (** ) set operator on is reversible clearly associated with the boundary operator a final operator in the sense of Kuratowskischen cover axioms, and so in connection with ( *).

This applies to the quantity system, so the amount of open sets of:

Set (mathematics) Interior (topology) Disjoint sets John L. Kelley Horst Schubert
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