What does it mean to have a "different topology"? On a space, I understand the notion of having different metrics on the same space. It is, in layman's terms, different ways of defining a distance but on the same space.
But I often see the term "different topology" being used, for example in this excellent answer. But I do not understand this idea so well.
What does it mean, essentially, to have a "different topology" on the same set say $\mathbb{R}$? Can you provide some simple examples that convey this idea?

A set $X$ can be equipped with what is called a topology. Then you get a topological space. The topology is a collection $T$ of subsets of $X$. The sets in this subset are exactly the sets that we called open. So, for example, you require that if $X$ and $Y$ are in $T$, then $X\cup Y$ is in $T$. All this is saying is that the union of two open sets is open. If you google "topological space" or "topology" you can find a list of the axioms needed to have a topology.
Saying that you can put a different topology on $X$ just means that you can pick another collection of subsets of $X$. For example, given any set $X$ you could let $T$ be the collection of all subsets of $X$. This is called the discrete topology on $X$. You can also always put the trivial topology on $X$. Here $T = \{\emptyset, X\}$. So only the empty set and the set itself is open.
Where do topologies come from then? One way they arise is when you have a metric (as discussed in the linked question/answer above). Different metrics simply can give different topologies.
