Why is the entire set $X$ open in every topology So I've just picked up topology and i read that for any set $X$ the entire set is an element of every topology and so is an open set.
but why is this so?
I understand that the complement of $X$ is $\emptyset$ which is open by vacuous truth however how do we then show that $X$ is open?
Thanks in advance.
 A: By definition of a topology the set $X$ and $\emptyset$ will lie in it, so that they are always both open and closed.
A: Given a set $X$, a topology is a collection of subsets of $X$ that satisfy some "axioms". Those axioms define what a topology is on $X$. One of those axioms says that both $X$ and the empty set are in the topology. This is how you "show" that $X$ is always open with respect to any topology.
Given any set $X$, it has the trivial topology which consists of only the empty set and the entire set $X$.

Notes.
It is a reasonable question to ask why those axioms, for instance why the whole space is always assumed to be open in the axioms. You may want to read this question:
Why the axioms for a topological space are those axioms?
A: It is possible to define topology by means of 'neighborhoods' in the first place.
Then a set $A$ is open if every point of $A$ has a neighborhood $B$ fully in $A$: $B\subseteq A$.
Since all these sets, including the neighborhoods (whatever they are) live inside $X$, the ambient set $X$ will satisfy this definition.
However, writing out the axioms for the neighborhood approach and working rigorously based on them would be more complicated than its cleaned up version that puts the open sets in the first place, and write up their basic properties as axioms.
