This may be a simple question, but I am nonetheless confused. I am just starting to learn about topology.
I am confused by part of the Wolfram Mathworld article on Topological Spaces. In it, it is said that a topological space is a collection of open subsets $T$ on a set $X$ has several essential characteristics.
Two of these make sense to me: the empty set $\phi$ is in $T$, and that $X$ is also in $T$. $\phi$ because $\phi$ is a subset of every set, and $X$ because $T$ is a collection of subsets of $X$.
However, it is then said:
The intersection of a finite number of sets in T is in T.
The union of an arbitrary number of sets in T is in T.
And that if we redefined $T$ as a collection of closed subsets, that, with the following changes, we would still have a topological space:
The intersection of an arbitrary number of sets in T is also in T.
The union of a finite number of sets in T is also in T.
I don't understand how the changes from a collection of open to closed subsets, along with the swapping between arbitrary and finite, give us a topological space. Any insight would be appreciated.