# Equivalence Of Definitions Of Connectedness

In Rudin's Analysis the definition of connectedness is given as follows: A subset $$A$$ of a metric space $$X$$ is said to be connected if $$A$$ cannot be written as $$A= C \cup D$$ where $$C$$ and $$D$$ are separated sets. But in Apostol's Analysis the definition of connectedness is given as : A metric space $$X$$ is said to be connected if it can not be written as $$X =A \cup B$$ where $$A$$ and $$B$$ are non empty disjoint open sets of $$X$$ I want to know that one definition is for subsets of metric spaces while the other is for metric spaces. Are these two definitions same or do they have different meaning.

They are equivalent for topological spaces, not just for metric spaces. If $$X=A\cup B$$, where $$A$$ and $$B$$ are non-empty disjoint open sets, then $$A$$ and $$B$$ are also closed, since each is the complement of an open set, so
$$A\cap\operatorname{cl}B=A\cap B=\varnothing=A\cap B=(\operatorname{cl}A)\cap B\,,$$
i.e., $$A$$ and $$B$$ are separated. Conversely, if $$A$$ and $$B$$ are separated, then $$A\cap\operatorname{cl}B=\varnothing$$, so $$B\subseteq\operatorname{cl}B\subseteq X\setminus A=B$$, and hence $$\operatorname{cl}B=B$$. Thus, $$B$$ is closed, and $$A$$ is therefore open. A similarly argument shows that $$B$$ is open.
The first definition is more convenient for subsets, but there is still a kind of equivalence: $$C$$ and $$D$$ are separated sets in $$X$$ if and only if they are relatively open subsets of $$A$$; they need not, however, be open in $$X$$ (and generally are not).
• Why $A$ and $B$ are also closed sets?? And what is $X$ is not a full space? – user728159 Oct 22 '20 at 18:18
• @user728159: The complement of an open set is always closed. Indeed, that is often taken as the definition of a closed set. And it is given that $X$ is the whole space. – Brian M. Scott Oct 22 '20 at 18:20
• But what if $X$ is not a full space?? – user728159 Oct 22 '20 at 18:20
• @user728159: I have no idea what you mean by a ‘full’ space; that is not any kind of standard terminology. A space is a space. Here $X$ is the space in which we’re working. – Brian M. Scott Oct 22 '20 at 18:21