4.1 Definition and Examples of Connected Spaces
Definition 4.1.1. A topological space \(X\) is said to be connected if the empty set \(\emptyset \) and the whole
set \(X\) are the only subsets of \(X\) that are both open and closed.
Example 4.1.2. If \(X=\{1,2,3\}\) and \(\tau =\{\{1\},\{2\},\{1,2\},\emptyset ,X\}\), then \((X,\tau )\) is a topological space. The open sets are \(\{1\}\), \(\{2\}\), \(\{1,2\}\), \(\emptyset \), \(X\). The closed sets
are \(\emptyset \), \(X\), \(\{2,3\}\), \(\{1,3\}\) and \(\{3\}\). The only sets that are both open and closed are \(\emptyset \) and \(X\). Hence \((X,\tau )\) is a connected
topological space.
The set of real numbers \(\mathbb {R}\) with the usual topology is connected. We have the following
characterization of connected topological spaces.
Theorem 4.1.3. A topological space \(X\) is connected if and only if it has the following property:
if \(U\) and \(V\) are non-empty open sets in \(X\) such that \(X=U\cup V\), then \(U\cap V\) is non-empty.
Proof. If \(U\) is a subset of \(X\) that is both closed and open and if \(V=X-U\), then \(U\) and \(V\) are both open, \(U\cup V=X\) and
\(U\cap V=\emptyset \).
Conversely, if \(U\) and \(V\) are open sets in \(X\) satisfying \(U\cup V=X\) and \(U\cap V=\emptyset \), then \(U=X-V\) and hence \(U\) is both open and closed.
Thus a topological space is connected if and only if there do not exists non-empty open sets \(U\)
and \(V\) such that \(U\cup V=X\) and \(U\cap V=\emptyset \). The result then follows.
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