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?
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.