Definition 2.1

Definition 2.1: Let $$X$$ be a set, and suppose $$T$$ is a collection of subsets of $$X$$ with the following properties:

i) $$\emptyset \in T$$;

ii) $$X \in T$$;

iii) The union of any collection of members of $$T$$ is also a member of $$T$$;

iv) The intersection of any two members of $$T$$ is also a member of $$T$$.

Then we say that the pair $$(X, T)$$ is a topological space and $$T$$ is the topology on $$X$$. (Often, when the topology $$T$$ is understood, we simply refer to the topological space as $$X$$.) The members of $$T$$ are called open sets.