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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.

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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).

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  • $\begingroup$ Why $A$ and $B$ are also closed sets?? And what is $X$ is not a full space? $\endgroup$ – user728159 Oct 22 '20 at 18:18
  • $\begingroup$ @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. $\endgroup$ – Brian M. Scott Oct 22 '20 at 18:20
  • $\begingroup$ But what if $X$ is not a full space?? $\endgroup$ – user728159 Oct 22 '20 at 18:20
  • $\begingroup$ @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. $\endgroup$ – Brian M. Scott Oct 22 '20 at 18:21

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