Edge (topology)

This article deals with the boundary of a subset of a topological space. For the boundary of a simplex see Simplex (Mathematics) # faces and boundary , for the boundary of a singular chain see chain complex # chain complex , for the boundary of a manifold see manifold with boundary .

An area (light blue) and its edge (dark blue).

In the mathematical sub-area of topology , the term edge is an abstraction of the descriptive representation of a boundary of an area.

definition

The edge of a subset of a topological space is the difference between the closure and interior of . The edge of a set is usually denoted by, so: ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle \partial U}$

(*) . ${\displaystyle \partial U={\overline {U}}\setminus U^{\circ }={\overline {U}}\cap {\overline {(X\setminus U)}}}$

The points from are called edge points . ${\displaystyle \partial U}$

Explanation

Every boundary point of is also a point of contact from and every point of contact is an element of or boundary point of . The points of contact of together form the conclusion of . So it is ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$

(**) ${\displaystyle {\overline {U}}=U\cup \partial U\,.}$

For each subset the topological space is divided into the interior of , the edge of and the exterior of : ${\displaystyle U\subseteq X}$ ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$

${\displaystyle X=U^{\circ }\;{\dot {\cup }}\;\partial U\;{\dot {\cup }}\;({X\setminus U})^{\circ }\,.}$

Demarcation

Both in algebraic topology and in the theory of bounded manifolds there are terms of “edge” that are related to the boundary concept of set theoretical topology discussed here, but do not agree with it (and with each other).

properties

• The edge of a set is always closed.
• The boundary of a set consists precisely of those points for which it is true that each of its surroundings contains both points from and points that are not in . ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$
• The edge of a set is always the same as the edge of its complement.
• The edge of a set is the intersection of the end of the set with the end of its complement.
• A set is closed exactly when it contains its edge.
• A set is open if and only if it is disjoint to its edge .
• A set is open and closed exactly when its margin is empty.
• Let there be a topological space, an open subset with the subspace topology and a subset. Then the edge of in is equal to the intersection of with the edge of in . If the requirement of openness is dropped, the corresponding statement generally does not apply, even if is. In the example , is also , and this set has no boundary at all, although it is identical to this one. ${\displaystyle X}$ ${\displaystyle Y\subseteq X}$ ${\displaystyle U\subseteq X}$ ${\displaystyle Y\cap U}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle U\subseteq Y}$ ${\displaystyle X=\mathbb {R} }$ ${\displaystyle U=Y=\{0\}}$ ${\displaystyle Y\cap U=\{0\}}$ ${\displaystyle Y=\{0\}}$ ${\displaystyle X}$

Examples

• If there is an open or closed circular disc in the plane , the edge of is the corresponding circular line . ${\displaystyle U}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle U}$
• The edge of as a subset of is whole . ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

Marginal axioms

For a topological space , the formation of the boundary is a set operator on , the power set of . This met for and remember to follow the four rules, the so-called edge axioms: ${\displaystyle X}$ ${\displaystyle {\mathcal {P}}(X)=\{U\mid U\subseteq X\}}$ ${\displaystyle X}$ ${\displaystyle U\subseteq X}$ ${\displaystyle V\subseteq X}$

(R1)   ${\displaystyle U\cap V\cap \partial (U\cap V)=U\cap V\cap (\partial U\cup \partial V)}$

(R2)   ${\displaystyle \partial U=\partial (X\setminus U)}$

(R3)   ${\displaystyle \partial {\partial U}\subseteq \partial U}$

(R4)   ${\displaystyle \partial {\emptyset }=\emptyset }$

The structure of the topological space is clearly defined by the four rules (R1) - (R4) . The set operator auf given by means of (**) is a closing operator in the sense of the Kuratovskian envelope axioms and thus, in connection with (*), is reversibly uniquely linked with the boundary operator . ${\displaystyle X}$ ${\displaystyle {\mathcal {P}}(X)}$ ${\displaystyle U\mapsto \partial U}$

The following applies to the set system , i.e. the set of open sets of : ${\displaystyle {\mathcal {\tau }}(X)}$ ${\displaystyle X}$

${\displaystyle {\mathcal {\tau }}(X)=\{U\subseteq X\mid {U\cap \partial U}=\emptyset \}}$

literature

• John L. Kelley : General topology (=  Graduate Texts in Mathematics . Volume 27 ). Springer, New York a. a. 1975, ISBN 3-540-90125-6 (reprint of the Van Nostrand, New York 1955 edition).
• Boto von Querenburg : Set theoretical topology . 3rd, revised and expanded edition. Springer, Berlin a. a. 2001, ISBN 3-540-67790-9 .
• Gerhard Preuss : General Topology . Springer, Berlin a. a. 1972, ISBN 3-540-06006-5 .
• Horst Schubert : Topology. An introduction . 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6 .
• Ramaswamy Vaidyanathaswamy : Set topology . 2nd ed., Reprinted. Chelsea Publishing, New York 1964.

Individual evidence

1. ^ Vaidyanathaswamy: Set topology. 1964, pp. 57-58.
2. ^ Schubert: Topology. 1975, p. 16.