Topology/Bases

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DefinitionEdit

Let (X,\mathcal{T}) be a topological space. A collection \mathcal{B} of open sets is called a base for the topology \mathcal{T} if every open set U is the union of sets in \mathcal{B}.

Obviously \mathcal{T} is a base for itself.

Conditions for Being a BaseEdit

In a topological space (X,\mathcal{T}) a collection \mathcal{B} is a base for \mathcal{T} if and only if it consists of open sets and for each point x\in X and open neighborhood U of x there is a set B\in\mathcal{B} such that x\in B\subseteq U.

Constructing Topologies from BasesEdit

Let X be any set and \mathcal{B} a collection of subsets of X. There exists a topology \mathcal{T} on X such that \mathcal{B} is a base for \mathcal{T} if and only if \mathcal{B} satisfies the following:

  1. If x\in X, then there exists a B\in\mathcal{B} such that x\in B.
  2. If B_1,B_2\in\mathcal{B} and x\in B_1\cap B_2, then there is a B\in\mathcal{B} such that x\in B\subseteq B_1\cap B_2.

Remark : Note that the first condition is equivalent to saying that The union of all sets in \mathcal{B} is X.

SemibasesEdit

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

ExercisesEdit

  1. Show that the collection \mathcal{B}=\{(a,b):a,b\in\mathbb{R},a<b\} of all open intervals in \mathbb{R} is a base for a topology on \mathbb{R}.
  2. Show that the collection \mathcal{C}=\{[a,b]:a,b\in\mathbb{R},a<b\} of all closed intervals in \mathbb{R} is not a base for a topology on \mathbb{R}.
  3. Show that the collection \mathcal{L}=\{(a,b]:a,b\in\mathbb{R},a<b\} of half open intervals is a base for a topology on \mathbb{R}.
  4. Show that the collection \mathcal{S}=\{[a,b):a,b\in\mathbb{R},a<b\} of half open intervals is a base for a topology on \mathbb{R}.
  5. Let a,b\in\mathbb{R}. A Partition \mathcal{P} over the closed interval [a,b]\,\! is defined as the ordered n-tuple a<x_1<x_2<\ldots <x_n<b \,\!; the norm of a partition \mathcal{P} is defined as \|\mathcal{P}\|=\sup \{x_i-x_{i-1}|2\leq i\leq n\}
    For every \delta >0\,\!, define the set U_{\delta}=\{\mathcal{P}|\|\mathcal{P}\|\leq\delta\}.
    If X\,\! is the set of all partitions on [a,b]\,\!, prove that the collection of all U_{\delta}\,\! is a Base over the Topology on X\,\!.


← Topological Spaces Bases Points in Sets →
Last modified on 11 April 2014, at 02:48