Hello! I am trying to understand the definition of topological space but I have some difficulties with it

Here it comes

Let $\displaystyle X$ be anyset and $\displaystyle T={U_{i}|i\in I}$ denote a certain collection of subsets of $\displaystyle X$. The pair $\displaystyle (X,T)$ is called a topological space if $\displaystyle T$ atisfies the following requirements:

(i) $\displaystyle \emptyset, X\in T$

(ii) if $\displaystyle T$ denotes any (maybe infinite) subcollection of $\displaystyle I$ , the family$\displaystyle {U_{j}|j\in J$ satifies $\displaystyle \cup_{j\in J}U_{j}\in T$.

(iii) if $\displaystyle K$ is any finite subcollection of $\displaystyle I$ the family $\displaystyle {U_{k}|k\in K$ satisfies $\displaystyle \cap_{k\in K}U_{K}\in T$

well I just do not get it.

1)Does it mean that [tex] T is some collection of susbsets $\displaystyle U_{i}$ so that the union of this subsets belongs to T (covers it?)

2)I understand that X has to belong to T but why even the empty set has to be there?

3) could someone give me descriptive interpretation of the conditions (ii) and (iii)

Can we just say that the first one requires the empty set and our set X to be in T

second one says that the union of any collections of sets in T is also in T

and the last one says that the intersection of any pair of sets in T is also in T?

But i still do not know why the empty set with X belongs to T???