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: 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.
A: 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.
A: 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.
A: Let's just look at $\Bbb R$ as an example. You're used to open sets being of the form $(a,b)$ for $a<b$, as well as unions of these types of sets. This is the standard topology on $\Bbb{R}$. 
What if we choose something else to be our open sets? We could choose sets of the form $[a,b)$ for $a<b$, for example, plus the unions of these sets. This is called the Sorgenfrey topology on $\Bbb{R}$. 
Okay, so what, we have different open sets now. Who cares? Well, it turns out that in topology we define continuity completely in terms of our open sets, so now the Sorgenfrey topology has a different notion of "continuity" than you're used to. Indeed, many properties of the "real line" change when we change our topology.
