Thread: Confusion in Topology definition

1. Confusion in Topology definition

In definition of topology, why we say intersection of finite members of τ belongs to τ. What is harm in any number of members of τ in it?

2. Re: Confusion in Topology definition

Originally Posted by makenqau1
In definition of topology, why we say intersection of finite members of τ belongs to τ. What is harm in any number of members of τ in it?
Think about $\mathbb{R}^2$ with the ordinary interval topology.
Now each of $O_n=\left(-1-n^{-1},1+n^{-1}\right)$ is a basic open set.
What is $\bigcap\limits_{n = 1}^\infty {{O_n}}~?$ Is the intersection in the topology?

3. Re: Confusion in Topology definition

Note that the "closed sets" (defined as the complements of the sets in the topology, the "open sets") have the property that the intersection of any number of closed sets is closed while the union of a finite collection of closed sets is closed. And it is quite possible to define a topology by giving its closed sets rather than its open sets.