What does "open set" mean in the concept of a topology?
Given the following definition of topology, I am confused about the concept of "open sets".
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 $r>0$ 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.
Solution 1:
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.
Solution 2:
In the topological space $( X, \mathcal T)$ a subset of $X$ is simply called open iff it is an element of $\mathcal T$. Nothing else. This terminology is justified by the fact that the set of open sets (by the definition that you know) in a metric space is a topology and that some of the theorems about open sets, continuity and so on carry over from metric spaces to topological spaces. There will surely be some examples shortly after the definition in your book.