Topology > Base

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Definition:

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 base:

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 .

Constructing topologies from base:

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 .

Semibasis:

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 .