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Section Summary
Important ideas that we discussed in this section include the following.
A topological space \((X,\tau)\) is connected if \(X\) cannot be written as the union of two disjoint, nonempty, open subsets. A subset \(A\) of a topological space topological space \((X,\tau)\) is connected if \(A\) is connected in the subspace topology.
A separation of a subset \(A\) of a topological space \(X\) is a pair of nonempty open subsets \(U\) and \(V\) of \(X\) such that
\(A \subseteq (U \cup V)\text{,}\)
\(U \cap A \neq \emptyset\text{,}\)
\(V \cap A \neq \emptyset\text{,}\) and
\(U \cap V \cap A = \emptyset\text{.}\)
Showing that a set has a separation can be a convenient way to show that the set is disconnected.
The connected subsets of \(\R\) are the intervals and the single point sets.
A subspace \(C\) of a topological space \(X\) is a connected component of \(X\) if \(C\) is connected and there is no larger connected subspace of \(X\) that contains \(C\text{.}\)
One application of connectedness is the Intermediate Value Theorem that tells us that if \(A\) is a connected subset of a topological space \(X\) and if \(f : A \to \R\) is a continuous function, then for any \(a,b \in A\) and any \(y \in \R\) between \(f(a)\) and \(f(b)\text{,}\) there is a point \(x \in A\) such that \(f(x) = y\text{.}\)
A subset \(S\) of a connected topological space \(X\) is a cut set of \(X\) if the set \(X \setminus S\) is disconnected, while a point \(p\) in \(X\) is a cut point if \(X \setminus \{p\}\) is disconnected. The property of being a cut set or a cut point is a topological invariant, so we can sometimes use cut sets and cut points to show that topological spaces are not homeomorphic.
