Basis for a topological space

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This article is about a basic definition in topology.
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Definition when the topological space is specified

Symbol-free definition

A basis for a topological space is a collection of open subsets of the topological space, such that every open subset can be expressed as a (possibly empty) union of basis subsets.

Definition with symbols

If X is a topological space, a basis for X is a collection \{ U_i \}_{i \in I} of open subsets of X (here, I is an indexing set) such that for any open subset V of X, there exists J \subset I such that:

V = \bigcup_{i \in J} U_j

Definition when the topological space is not specified

Symbol-free definition

Given a set, a collection of subsets of the set is said to form a basis for a topological space or a basis for a topology if the following two conditions are satisfied:

  • The union of all members of the collection is the whole space
  • Any finite intersection of members of the collection, is itself a union of members of the collection

The topology generated by this basis is the topology in which the open sets are precisely the unions of basis sets.

Definition with symbols

In symbols: if X is a set, a collection \{ U_i \}_{i \in I} of subsets of X is said to form a basis for a topology on X if the following two conditions are satisfied:

  • \bigcup_{i \in I} U_i = X
  • For all i, j \in I, there exists J \subset I such that:

U_i \cap U_j = \bigcap_{k \in J} U_k

The second condition is sometimes stated as follows: if p \in U_i \cap U_j, then there exists U_k \ni p such that U_k \subset U_i \cap U_j.

The topology generated by the U_is is defined as follows: a subset V is open in X if and only if there exists

Equivalence of definitions

Further information: Equivalence of definitions of basis

Related notions

Examples

Extreme examples

  1. For any topological space, the collection of all open subsets is a basis. That's because any open subset of a topological space can be expressed as a union of size one.
  2. For a discrete topological space, the collection of one-point subsets forms a basis. That's because every open subset of a discrete topological space is a union of one-point subsets, namely, the one-point subsets corresponding to its elements.

Examples from metric spaces

  1. Euclidean space: A basis for the usual topology on Euclidean space is the open balls. An open ball of radius r > 0 centered at a point x, is defined as the set of all y \in \R^n whose distance from x is strictly smaller than r. By the way the topology on \R^n is defined, these open balls clearly form a basis.
  2. Metric space: Given any metric space, there is a natural way of viewing it as a topological space. This natural way involves declaring the collection of open balls in the metric space as a basis. To see that this gives a well-defined topology, we need to check that the collection of open balls satisfies the conditions to be a basis for a topological space. This follows from the conditions for a metric space, including nonnegativity, symmetry and the triangle inequality.