Topology/Points in Sets

Topology
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Some Important ConstructionsEdit

Let   be a topological space and   be any subset of  .

ClosureEdit

  • A point   is called a point of closure of   if every neighborhood of   contains at least one element of  . In other words, for all neighborhoods   of  ,  .
  • The closure of   is the set of all points of closure of  . It is equivalent to the intersection of all closed sets that contain   as a subset, denoted   (some authors use  ). Alternatively, it is the set   together with all its limit points (defined below). The closure has the nice property of being the smallest closed set containing  . All neighborhoods of each point in the closure intersects  .

InteriorEdit

  • A point   is an internal point of   if there is an open subset of   containing  .
  • The interior of   is the union of all open sets contained inside  , denoted   (some authors use  ). The interior has the nice property of being the largest open set contained inside  . Every point in the interior has a neighborhood contained inside  . It is equivalent to the set of all interior points of  .

Note that an open set is equal to its interior.

ExteriorEdit

  • Define the exterior of   to be the union of all open sets contained inside the complement of  , denoted  . It is the largest open set inside  . Every point in the exterior has a neighborhood contained inside  .

BoundaryEdit

  • Define the boundary of   to be the closure of   excluding its interior, or  . It is denoted   (some authors prefer  ). The boundary is also called the frontier. It is always closed since it is the intersection of the closed set   and the closed set  . It can be proved that   is closed if it contains all its boundary, and is open if it contains none of its boundary. Every neighborhood of each point in the boundary intersects both   and  . All boundary points of a set   are obviously points of contact of  .

Limit PointsEdit

  • A point   is called a limit point of   if every neighborhood of   intersects   in at least one point other than  . In other words, for every neighborhood   of  ,  . All limit points of   are obviously points of closure of  .

Isolated PointsEdit

  • A point   of   is an isolated point of   if it has a neighborhood which does not contain any other points of  . This is equivalent to saying that   is an open set in the topological space   (considered as a subspace of  ).

DensityEdit

Definition:   is called dense (or dense in  ) if every point in   either belongs to   or is a limit point of  . Informally, every point of   is either in   or arbitrarily close to a member of  . For instance, the rational numbers are dense in the real numbers because every real number is either a rational number or has a rational number arbitrarily close to it.

Equivalently:   is dense if the closure of   is  .

Definition:   is nowhere dense (or nowhere dense in  ) if the closure of   has an empty interior. That is, the closure of   contains no non-empty open sets. Informally, it is a set whose points are not tightly clustered anywhere. For instance, the set of integers is nowhere dense in the set of real numbers. Note that the order of operations matters: the set of rational numbers has an interior with empty closure, but it is not nowhere dense; in fact it is dense in the real numbers.

Definition: A Gσ set is a subset of a topological space that is a countable intersection of open sets.

Definition: An Fσ set is a countable union of closed sets.

Theorem

(Hausdorff Criterion) Suppose X has 2 topologies, r1 and r2. For each  , let B1x be a neighbourhood base for x in topology r1 and B2x be a neighbourhood base for x in topology r2. Then,   if and only if at each  , if  

Theorem

In any topological space, the boundary of an open set is closed and nowhere dense.

Proof:
Let A be an open set in a topological space X. Since A is open, int(A) = A. Thus, σA ( or the boundary of A) =  . Note that  . The complement of an open set is closed, and the closure of any set is closed. Thus,   is an intersection of closed sets and is itself closed. A subset of a topological space is nowhere dense if and only if the interior of its closure is empty. So, proceeding in consideration of the boundary of A.

The interior of the closure of the boundary of A is equal to the interior of the boundary of A.
Thus, it is equal to  .
Which is also equal to  .

And,  . So, the interior of the closure of the boundary of A =  ., and as such, the boundary of A is nowhere dense.

Types of SpacesEdit

We can also categorize spaces based on what kinds of points they have.

Perfect SpacesEdit

  • If a space contains no isolated points, then the space is a perfect space.

Some Basic ResultsEdit

  • For every set  ;   and  
    Proof:
    Let  . If a closed set  , then  . As   for closed  ; we have  .   being arbitrary,  
    Let   be open. Thus,  . As   for open  ; we have  .   being arbitrary, we have  


  • A set   is open if and only if  .
    Proof:
    ( )
      is open and  . Hence,  . But we know that   and hence  
    ( )
    As   is a union of open sets, it is open (from definition of open set). Hence   is also open.


  • A set   is closed if and only if  
    Proof:
    Observe that the complement of   satisfies  . Hence, the required result is equivalent to the statement "  is open if and only if  ".   is closed implies that   is open, and hence we can use the previous property.


  • The closure   of a set   is closed
    Proof:
    Let   be a closed set such that  . Now,   for closed  . We know that the intersection of any collection of closed sets is closed, and hence   is closed.

ExercisesEdit

  1. Prove the following identities for subsets   of a topological space  :
    •  
    •  
    •  
    •  
  2. Show that the following identities need not hold (i.e. give an example of a topological space and sets   and   for which they fail):
    •  
    •  


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