Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have seen two different definitions of disconnected sets. Here (X,T) is a topological space.

A subset $A$ of X is disconnected if there are two open sets $U,V$ of X such that $U \cap A \neq \emptyset $ , $V \cap A \neq \emptyset $, $U \cap A \cap V = \emptyset $ and $A\subset U\cup V$.


Two subsets $A,B$ of X are separated if no point of A lies in the closure of B and vice versa. A subset E of X is disconnected if E is a union of two nonempty separated sets.

I am having trouble understanding the connection between these two definitons. Why do they mean the same thing?

Thank you kindly.

share|cite|improve this question
How can they mean the same thing if the first is about a single set $A$, and the second is about a pair of sets $A,B$? – Thomas Andrews Dec 12 '11 at 14:49
First is the definition of A is disconnected and expressed in terms of two open sets U,V. Second is the definiton of E is disconnected and expressed in terms of A,B two seperated sets. – marvinthemartian Dec 12 '11 at 14:54
Whoops, reading comprehension error. – Thomas Andrews Dec 12 '11 at 15:13
up vote 3 down vote accepted

In your second definition of separated, suppose that $E$ is a union of two non-empty separated sets $A$ and $B$. Define $U$ to be $X-\overline{B}$ and $V=X-\overline{A}$, where $\overline{A}$ and $\overline{B}$ are the topological closures of $A$ and $B$, respectively. Then, $U$ and $V$ are open sets and if $A$ and $B$ are separated, then $A\subseteq U$ and $B\subseteq V$. Since $E=A\cup B$, we see that $U\cap E\neq \emptyset$, $V\cap E\neq \emptyset$, but $U\cap E\cap V =\emptyset$ and $E=A\cup B\subseteq U\cup V$.

share|cite|improve this answer
Minor typo: should be $A \subseteq U$ and $B \subseteq V$ (not $B \subseteq A$). – mjqxxxx Dec 12 '11 at 15:13
Thanks! Fixed the typo. – Álvaro Lozano-Robledo Dec 12 '11 at 15:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.