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 $X$ is a collection of points. By itself, it does not have any topology associated to it. A topology on $X$ is a collection $T$ of subsets of $X$, called open sets, satisfying a number of axioms, and a topological space is a pair $(X,T)$ where $X$ is some set and $T$ is a topology on $X$.
Given some set $X$, it generally possible to construct multiple different collections $T$ of subsets of $X$ satisfying the axioms. For example if $X = \{1,2\}$, then the following collections work:


*

*$T_1 = \{ \varnothing, \{1,2\} \}$;

*$T_2 = \{ \varnothing, \{1\}, \{1,2\} \}$;

*$T_3 = \{ \varnothing, \{2\}, \{1,2\} \}$;

*$T_4 = \{ \varnothing, \{1\}, \{2\}, \{1,2\} \}$.


These are different collections of subsets of $X$ satisfying the axiom of a topology on $X = \{1,2\}$, so we just say they are different topologies.
Often some sets have a topology that comes naturally with them, or that is so often used in conjunction with the set that we don't bother mentioning it. For example with $\mathbb{R}$, the topoplogy coming from the metric is so often used that one usually doesn't even mention it. But formally it's still there, and by itself $\mathbb{R}$ is just a set like any other; if you can construct another collection of subsets of $\mathbb{R}$ satisfying the axioms of a topology, then you can say you've constructed another topology on $\mathbb{R}$, which may have nothing to do with the usual topology.
