Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of not belonging to the interior of An element of the boundary of is called a boundary point of The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include and Some authors (for example Willard, in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary.[1] Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.[2]

A set (in light blue) and its boundary (in dark blue).

A connected component of the boundary of is called a boundary component of

Common definitionsEdit

There are several equivalent definitions for the boundary of a subset   of a topological space   which will be denoted by     or simply   if   is understood:

  1. It is the closure of   minus the interior of   in  :
     
    where   denotes the closure of   in   and   denotes the topological interior of   in  
  2. It is the intersection of the closure of   with the closure of its complement:
     
  3. It is the set of points   such that every neighborhood of   contains at least one point of   and at least one point not of  :
     

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.

PropertiesEdit

The closure of a set   equals the union of the set with its boundary:

 
where   denotes the closure of   in   A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary. The boundary of a set is closed;[3] this follows from the formula   which expresses   as the intersection of two closed subsets of  

("Trichotomy") Given any subset   each point of   lies in exactly one of the three sets   and   Said differently,

 
and these three sets are pairwise disjoint. Consequently, if these set are not empty[note 1] then they form a partition of  

A point   is a boundary point of a set if and only if every neighborhood of   contains at least one point in the set and at least one point not in the set. The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set.

 
Conceptual Venn diagram showing the relationships among different points of a subset   of     = set of limit points of     set of boundary points of   area shaded green = set of interior points of   area shaded yellow = set of isolated points of   areas shaded black = empty sets. Every point of   is either an interior point or a boundary point. Also, every point of   is either an accumulation point or an isolated point. Likewise, every boundary point of   is either an accumulation point or an isolated point. Isolated points are always boundary points.

ExamplesEdit

Characterizations and general examplesEdit

The boundary of a set is equal to the boundary of the set's complement:

 

If   is a dense open subset of   then  

The interior of the boundary of a closed set is the empty set.[proof 1] Consequently, the interior of the boundary of the closure of a set is the empty set. The interior of the boundary of an open set is also the empty set.[proof 2] Consequently, the interior of the boundary of the interior of a set is the empty set. In particular, if   is a closed or open subset of   then there does not exist any non-empty subset   such that   is also an open subset of   This fact is important for the definition and use of nowhere dense subsets, meager subsets, and Baire spaces.

A set is the boundary of some open set if and only if it is closed and nowhere dense. The boundary of a set is empty if and only if the set is both closed and open (that is, a clopen set).

Concrete examplesEdit

 
Boundary of hyperbolic components of Mandelbrot set

Consider the real line   with the usual topology (that is, the topology whose basis sets are open intervals) and   the subset of rational numbers (whose topological interior in   is empty). Then

  •  
  •  
  •  
  •  

These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. They also show that it is possible for the boundary   of a subset   to contain a non-empty open subset of  ; that is, for the interior of   in   to be non-empty. However, a closed subset's boundary always has an empty interior.

In the space of rational numbers with the usual topology (the subspace topology of  ), the boundary of   where   is irrational, is empty.

The boundary of a set is a topological notion and may change if one changes the topology. For example, given the usual topology on   the boundary of a closed disk   is the disk's surrounding circle:   If the disk is viewed as a set in   with its own usual topology, that is,   then the boundary of the disk is the disk itself:   If the disk is viewed as its own topological space (with the subspace topology of  ), then the boundary of the disk is empty.

Boundary of an open ball vs. its surrounding sphereEdit

This example demonstrates that the topological boundary of an open ball of radius   is not necessarily equal to the corresponding sphere of radius   (centered at the same point); it also shows that the closure of an open ball of radius   is not necessarily equal to the closed ball of radius   (again centered at the same point). Denote the usual Euclidean metric on   by

 
which induces on   the usual Euclidean topology. Let   denote the union of the  -axis   with the unit circle   centered at the origin  ; that is,   which is a topological subspace of   whose topology is equal to that induced by the (restriction of) the metric   In particular, the sets   and   are all closed subsets of   and thus also closed subsets of its subspace   Henceforth, unless it clearly indicated otherwise, every open ball, closed ball, and sphere should be assumed to be centered at the origin   and moreover, only the metric space   will be considered (and not its superspace  ); this being a path-connected and locally path-connected complete metric space.

Denote the open ball of radius   in   by   so that when   then

 
is the open sub-interval of the  -axis strictly between   and   The unit sphere in   ("unit" meaning that its radius is  ) is
 
while the closed unit ball in   is the union of the open unit ball and the unit sphere centered at this same point:
 

However, the topological boundary   and topological closure   in   of the open unit ball   are:

 
In particular, the open unit ball's topological boundary   is a proper subset of the unit sphere   in   And the open unit ball's topological closure   is a proper subset of the closed unit ball   in   The point   for instance, cannot belong to   because there does not exist a sequence in   that converges to it; the same reasoning generalizes to also explain why no point in   outside of the closed sub-interval   belongs to   Because the topological boundary of the set   is always a subset of  's closure, it follows that   must also be a subset of  

In any metric space   the topological boundary in   of an open ball of radius   centered at a point   is always a subset of the sphere of radius   centered at that same point  ; that is,

 
always holds.

Moreover, the unit sphere in   contains   which is an open subset of  [proof 3] This shows, in particular, that the unit sphere   in   contains a non-empty open subset of  

Boundary of a boundaryEdit

For any set   where   denotes the superset with equality holding if and only if the boundary of   has no interior points, which will be the case for example if   is either closed or open. Since the boundary of a set is closed,   for any set   The boundary operator thus satisfies a weakened kind of idempotence.

In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the singular homology rests critically on this fact. The explanation for the apparent incongruity is that the topological boundary (the subject of this article) is a slightly different concept from the boundary of a manifold or of a simplicial complex. For example, the boundary of an open disk viewed as a manifold is empty, as is its topological boundary viewed as a subset of itself, while its topological boundary viewed as a subset of the real plane is the circle surrounding the disk. Conversely, the boundary of a closed disk viewed as a manifold is the bounding circle, as is its topological boundary viewed as a subset of the real plane, while its topological boundary viewed as a subset of itself is empty. In particular, the topological boundary depends on the ambient space, while the boundary of a manifold is invariant.

See alsoEdit

NotesEdit

  1. ^ The condition that these sets be non-empty is needed because sets in a partition are by definition required to be non-empty.
  1. ^ Let   be a closed subset of   so that   and thus also   If   is an open subset of   such that   then   (because  ) so that   (because by definition,   is the largest open subset of   contained in  ). But   implies that   Thus   is simultaneously a subset of   and disjoint from   which is only possible if    
  2. ^ Let   be an open subset of   so that   Let   so that   which implies that   If   then pick   so that   Because   is an open neighborhood of   in   and   the definition of the topological closure   implies that   which is a contradiction.   Alternatively, if   is open in   then   is closed in   so that by using the general formula   and the fact that the interior of the boundary of a closed set (such as  ) is empty, it follows that    
  3. ^ The  -axis   is closed in   because it is a product of two closed subsets of   Consequently,   is an open subset of   Because   has the subspace topology induced by   the intersection   is an open subset of    

CitationsEdit

  1. ^ Hausdorff, Felix (1914). Grundzüge der Mengenlehre. Leipzig: Veit. p. 214. ISBN 978-0-8284-0061-9. Reprinted by Chelsea in 1949.
  2. ^ Hausdorff, Felix (1914). Grundzüge der Mengenlehre. Leipzig: Veit. p. 281. ISBN 978-0-8284-0061-9. Reprinted by Chelsea in 1949.
  3. ^ Mendelson, Bert (1990) [1975]. Introduction to Topology (Third ed.). Dover. p. 86. ISBN 0-486-66352-3. Corollary 4.15 For each subset     is closed.

ReferencesEdit