# Cut

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in a domain $D \subset \mathbf C$ along a non-closed simple arc $\gamma = \{ {z ( t) } : {0 \leq t \leq 1 } \}$

The removal of the points of the arc $\gamma$ from the domain $D$, that is, replacing the domain $D$ by the domain (or domains) $D \setminus \gamma$, as well as the set $\gamma$ itself. Here it is assumed that either the whole arc $\gamma$ or the whole arc except the initial or end point $z ( 0), z ( 1)$ belongs to $D$ and that $z ( 0)$ or $z ( 1)$ belong to the boundary $\partial D$. To each point $z ( t)$ of the cut $\gamma$, when $0 < t < 1$, there correspond two prime ends of the part of the domain $D$ which belongs to $\gamma$; the left and right prime ends (cf. Limit elements). The union of these prime ends form the left and right-hand sides of the cut $\gamma$.

#### Comments

A cut is also called a slit.

One also speaks of boundary elements instead of limit elements or prime ends. These notions do not coincide in general, but for "nice" domains $D$( e.g. with Jordan boundary) they can be identified. A related notion is that of a crosscut: an open simple arc that begins and ends at two different points of $\partial D$, cf. [a1], especially Chapt. 3.

The word "cut" also occurs in several more meanings in various parts of mathematics. Thus, there is the notion of a Dedekind cut in the real or rational numbers, cf. Real number. The notion cut point on a geodesic line; and a cut or cutset in a graph or transportation network (cf. Flow in a network for the latter). Finally there are the cut locus and the cutting problem.

#### References

 [a1] M. Ohtsuka, "Dirichlet problem, extremal length and prime ends" , v. Nostrand-Reinhold (1970)
How to Cite This Entry:
Cut. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cut&oldid=46567
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article