Conflicting Definition for Separated Sets

Question

I'm comparing two definitions of separated sets,

1. Rudin, Principles of Mathematical Analysis (p.42) Two subsets $A$ and $B$ of a metric space $X$ are separated if,
   $$
   A \cap \overline{B} = \emptyset \quad \text{and} \quad \overline{B} \cap A = \emptyset
   $$

2. Munkres, Topology (p.148) Let $X$ be a topological space. A separation of $X$ is a pair of disjoint $U,V$ non-empty open subsets of $X$ whose union is $X$

I don't understand why these definitions aren't the same (or if they are, could someone pls explain the similarity)? Do the different contexts (metric vs topological spaces) motive the reason for definition similar objects differently, and if so, why?

I'm new to these subjects — so if I am missing something simple, thanks for helping me out 🙂

Answer 1

I don't think the different contexts are too relevant here. Both definitions only use that $X$ is a topological space.

However, the definitions define two different things:

1. This defines what it means for two subsets of $X$ to be separated from each other.
2. This defines what it means for two subsets of $X$ to be a separation of $X$

The two definitions are related though: An equivalent statement to 2. is: A separation of $X$ is a pair of non-empty separated subsets (in the sense of definition 1) such that their union is $X$.