TITLE: Definition of topological space: Is Ω equal to the powerset of X?
QUESTION [12 upvotes]: A topological space is a set $X$ and a collection $\Omega$ of subsets of $X$ such that:

$\emptyset \in \Omega$ and $X \in \Omega$
The union of any collection of $\Omega$ is in $\Omega$
The intersection of any finite collection of sets in $\Omega$ is again in $\Omega$

My question is, is the set $\Omega$ essentially the powerset of $X$? Or is the powerset of $X$ just a special case of a suitable collection of subsets of $X$?

REPLY [2 votes]: Let X = {a,b} be a set with 2 elements. There are four distinct topologies on X:
{∅, {a,b}} (the trivial topology)
{∅, {a}, {a,b}}
{∅, {b}, {a,b}}
{∅, {a}, {b}, {a,b}} (the discrete topology)

(wikipedia: Finite topological space)
The second and third of these are topologies, but are not power sets of X.
Afaict if all the elements of X are elements of the topology then there is only one topology, the discreet, identical to the powerset of X.