Definition 18.9.
A subspace \(C\) of a topological space \(X\) is a component (or connected component) of \(X\) if \(C\) is connected and there is no larger connected subspace of \(X\) that contains \(C\text{.}\)
Section Components
As Activity 18.6 demonstrates, spaces like \(A = (1,2) \cup (3,4)\) are not connected. Even so, \(A\) is made of two connected subsets \((1,2)\) and \((3,4)\text{.}\) These connected subsets are called components.
As an example, if \(X = (1,2) \cup [4,10) \cup \{-1,15\}\text{,}\) then the components of \(X\) are \((1,2)\text{,}\) \([4,10)\text{,}\) \(\{-1\}\) and \(\{15\}\text{.}\) As the next activity shows, we can always partition a topological space into a disjoint union of compenents.
Activity 18.7.
Let \((X, \tau)\) be a nonempty topological space.
(a)
Show that if \(x \in X\text{,}\) then \(\{x\}\) is connected.
(b)
Suppose that \(X\) is a topological space and \(\{A_{\alpha}\}\) for \(\alpha\) in some indexing set \(I\) is a collection of connected subsets of \(X\text{.}\) Let \(A = \bigcup_{\alpha \in I} A_{\alpha}\text{.}\) Suppose that \(\bigcap_{\alpha \in I} A_{\alpha} \neq \emptyset\text{.}\) Show that \(A\) is a connected subset of \(X\text{.}\)
Hint.
Assume a separation and use Lemma 18.7.
(c)
Part (a) shows that every element in \(x\) belongs to some connected subset of \(X\text{.}\) So we can write \(X\) as a union of connected subsets. But there is probably overlap. To remove the overlap, we define the following relation \(\sim\) on \(X\text{:}\) For \(x\) and \(y\) in \(X\text{,}\) \(x \sim y\) if \(x\) and \(y\) are contained in the same connected subset of \(X\text{.}\) As with any relation, we ask if \(\sim\) is an equivalence relation.
(i)
Is \(\sim\) a reflexive relation? Why or why not?
(ii)
Is \(\sim\) a symmetric relation? Why or why not?
(iii)
Is \(\sim\) a transitive relation? Why or why not?
Activity 18.7 shows that the relation \(\sim\) is an equivalence relation, and so partitions the underlying topological space \(X\) into disjoint sets. If \(x \in X\text{,}\) then the equivalence class of \(x\) is a connected subset of \(X\text{.}\) There can be no larger connected subset of \(X\) that contains \(x\text{,}\) since equivalence classes are disjoint or the same. So the equivalence classes are exactly the connected components of \(X\text{.}\) The components of a topological space \(X\) satisfy several properties.
- Each \(a \in X\) is an element of exactly one connected component \(C_a\) of \(X\text{.}\)
- A component \(C_a\) contains all connected subsets of \(X\) that contain \(a\text{.}\) Thus, \(C_a\) is the union of all connected subsets of \(X\) that contain \(a\text{.}\)
- If \(a\) and \(b\) are in \(X\text{,}\) then either \(C_a = C_b\) or \(C_a \cap C_b = \emptyset\text{.}\)
- Every connected subset of \(X\) is a subset of a connected component.
- Every connected component of \(X\) is a closed subset of \(X\text{.}\)
- The space \(X\) is connected if and only if \(X\) has exactly one connected component.
