Topology/Bases

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Contents

DefinitionEdit

Let   be a topological space. A collection   of open sets is called a base for the topology   if every open set   is the union of sets in  .

Obviously   is a base for itself.

Conditions for Being a BaseEdit

In a topological space   a collection   is a base for   if and only if it consists of open sets and for each point   and open neighborhood   of   there is a set   such that  .

Proof:
We need to show that a subset   of   is open if and only if it is is a union of elements in  . However, the if part is obvious, from the facts that the elements in   are open, and that so are arbitrary unions of open sets. Thus, we only have to prove, that any open set   indeed is such a union.
Let   be any open set. Consider any element  . By assumption, there is at least one element in  , which both contains   and is a subset of  . By the axiom of choice, we may simultaneously for each   choose such an element  . The union of all of them indeed is  . Thus, any open set can be formed as a union of sets within  .

Constructing Topologies from BasesEdit

Let   be any set and   a collection of subsets of  . There exists a topology   on   such that   is a base for   if and only if   satisfies the following:

  1. If  , then there exists a   such that  .
  2. If   and  , then there is a   such that  .

Remark : Note that the first condition is equivalent to saying that The union of all sets in   is  .

SemibasesEdit

Let   be any set and   a collection of subsets of  . Then   is a semibase if a base of X can be formed by a finite intersection of elements of  .

ExercisesEdit

  1. Show that the collection   of all open intervals in   is a base for a topology on  .
  2. Show that the collection   of all closed intervals in   is not a base for a topology on  .
  3. Show that the collection   of half open intervals is a base for a topology on  .
  4. Show that the collection   of half open intervals is a base for a topology on  .
  5. Let  . A Partition   over the closed interval   is defined as the ordered n-tuple  ; the norm of a partition   is defined as  
    For every  , define the set  .
    If   is the set of all partitions on  , prove that the collection of all   is a Base over the Topology on  .


Topology
 ← Topological Spaces Bases Points in Sets →