Why are there two definitions of "open"?

Question

I am learning topology, and I don't understand about an open set.

So I learned a definition of open set.

A subset U of M is called open (with respect to d) if for every x ∈ U there exists a real number ε > 0 so that B(x,ε)⊆U. (B is an open ball).

However, I've also learned that...

Let (X,T) be a topological space. If U⊆X, then U is open.

I failed to see the connection between those two definitions of "open". Do those two definitions actually define the same "open"? If so, can you explain? Or did I miss something?

Answer 1

First, something is wrong with your second definition (it says that any subset of a topological space is open). What I think you mean is that if U⊆T, then U is open. This definition doesn't really say much about open sets, but really just states that a topology on X is just a prescription of which sets are open. A topology T of X is a collection of subsets of X with the following properties:

1. X and ø are members of T
2. Arbitrary unions of members of T are also members of T
3. Finite intersections of members of T are also members of T

The members of T (which are subsets of X) are defined to be the open subsets of X.

You'll find several definitions because certain topological spaces have more structure than others. For example, your definition about using ε-balls only makes sense for a metric space (otherwise, being "ε away" doesn't mean anything).