A description/proof of that topological space is connected iff its open and closed subsets are only it and empty set
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of connected topological space.
- The reader knows a definition of closed set.
Target Context
- The reader will have a description and a proof of the proposition that any topological space is connected if and only if its open and closed subsets are only the topological space and the empty set.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
Any topological space, \(T\), is connected if and only if its open and closed subsets are only \(T\) and \(\emptyset\).
2: Proof
Suppose that the open and close subsets of \(T\) are only \(T\) and \(\emptyset\). Suppose that \(T\) was not connected. \(T = U_1 \cup U_2\), \(U_1 \cap U_2 = \emptyset\) where \(U_i\) would be a non-empty open set on \(T\). \(T \setminus U_1 = U_2\) would be closed, so, \(U_2\) would be an open and closed subset, a contradiction, so, \(T\) is connected.
Suppose that \(T\) is connected. \(T\) and \(\emptyset\) are open and closed subsets. Suppose that there was another open and closed subset, \(U \subseteq T\). \(T \setminus U\) would be open and non-empty, \(U \cap (T \setminus U) = \emptyset\), and \(T = U \cup (T \setminus U)\), so, \(T\) would not be connected, a contradiction, so, \(T\) and \(\emptyset\) are the only open and closed subsets.
