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?

The topology of a set is essentially a notion about the shape of the set. If you have a topology on a set, you can talk about neighbours, convergence, etc. There are many ways to give a topological structure to a set, as you know, for example taking different metrics may lead to different topologies. For example, the sets $X_1 =\mathbb{N}$ and $X_2 = \{\frac{1}{n}:n\in\mathbb{N} \}\cup\{ 0 \}$ are indistinguishable from the point of view of set theory. On the other hand, as subsets of $\mathbb{R}$, they have a topology induced by the usual metric. Topologically they have a different structure. $X_2$ has a distinguished point, namely, $0$, since every open set $U$ in $\mathbb{R}$ containing $0$ also contains infinitely many other elements of $X_2$. On the other hand, there are no points in $X_1$ having the same property. This is a common example where two set having the same set theoretical properties have different topological properties.
