Basis and Subbasis

Basis and Subbasis

Basis for a Topology

Topology Generated by a Basis

Equivalently, is open iff it is a union of basis elements.

Examples

Example 1: Standard basis for :

Example 2: Standard basis for : open balls

Example 3: generates the lower limit topology on (strictly finer than standard).

Comparing Topologies from Bases

Subbasis

The topology generated by has basis consisting of all finite intersections of elements of .

Subbasis for Product Topology

Neighborhood Basis

Second Countable Spaces

First Countable Spaces

Basis for the Product Topology

For finite products, the collection is a basis.

For infinite products, we use the subbasis described above (only finitely many proper projections).

Important Bases

• Metric topology: Basis of open balls
• Order topology: Basis of open intervals
• Product topology: Basis of products of open sets
• Subspace topology: