What does "open set" mean in the concept of a topology? Given the following definition of topology, I am confused about the concept of &quot;open sets&quot;.

2.2 Topological Space. We use some of the properties of open sets in the case of metric spaces in order to define what is meant in general by a class of open sets and by a topology.
Definition 2.2. Let $X$ be a nonempty set. A topology $\mathcal{T}$ for $X$ is a collection of subsets of $X$ such that $\emptyset,X\in\mathcal{T}$, and $\mathcal{T}$ is closed under arbitrary unions and finite intersections.
    We say $(X,\mathcal{T})$ is a topological space. Members of $\mathcal{T}$ are called open sets.
    If $x\in X$ then a neighbourhood of $x$ is an open set containing $x$.

It seems to me that the definition of an open subset is that subset $A$ of a metric space $X$ is called open if for every point $x \in A$ there exists $r&gt;0$ such that $B_r(x)\subseteq A$. What is the difference of being open in a metric space and being open in a topological space?
Thanks so much.
In an abstract topological space, "open set" has no definition!

You simply decide (as part of making your topological space) which sets you want to call open -- those are the sets you put into $\mathcal T$. Whatever you decide to call open will be called open, as long as your decision meets the condition "$\emptyset, X\in\mathcal T$ and $\mathcal T$ is closed under arbitrary unions and finite intersections".

A metric space becomes a topological space by deciding that the "open sets" in this particular topological space are going to be exactly the ones that are open according to the metric-space definition.
Short answer: It's a generalization of the concept of an open interval on the real line to arbitrary spaces. 

You'll see statements that the concept of 'open set' has no special meaning. That's only true in the most abstract sense. Really, it's just a convention.

Point-set topology is usually developed from the concept of 'open set' where a topology is defined by defining what its open sets are, and the topology's properties develop from that. (A metric space, of course, defines its open sets from the metric.)

However, the Kuratowski closure theorems show that we can just as easily start by defining the closed sets. It doesn't gain us anything but it's interesting to know as we go along with convention and proceed by defining the open sets first.

Topology developed as a way to understand problems of real analysis, and of course we always try relate any abstract concept of topology back to the Euclidean topology $E^n$ of $\mathbb{R}^n$ (sometimes called the 'usual topology' to emphasize the abstraction).

What are the open sets in $E$? Collections of open intervals. Everything else proceeds from that.