Section Introduction
The term “connected” should bring up images of something that is one piece, not separated. There is more than one way we can interpret the notion of connectedness in topological spaces. For example, we might consider a topological space to be connected if we can’t separate it into disjoint pieces in any non-trivial way. As another possibility, we might consider a topological space to be connected if there is always a path from one point in the spaced to another, provided we define what “path” means. These are different notions of connectedness, and we focus on the first notion in this section.
Connectedness is an important property, and one that we encounter in the calculus. For example, we will see in this section that the Intermediate Value Theorem relies on connected subsets of \(\R\text{.}\) To define a connected set, we will need to have a way to understand when and how a set can be separated into different pieces. Since a topology is defined by open sets, when we want to separate objects we will do so with open sets. This is similar to the idea behind Hausdorff spaces, except that we now want to know if a set can be separated in some way rather than separating points.
As an example to motivate the definition, consider the sets \(X = (0,1) \cup (1,2)\) and \(Y = [1,2]\) in \(\R\) with the Euclidean topology. Notice that we can write \(X\) as the union of two disjoint open sets \(X_1 = (0,1)\) and \(X_2 = (1,2)\text{.}\) So we shouldn’t think of \(X\) as being connected. However, if we attempt to write \(Y\) as a union of two subsets, say \(Y_1 = [1,1.5)\) and \(Y_2 = [1.5,2]\text{,}\) it is impossible for both of these subsets to be open and disjoint. So \(Y\) is a set we should consider to be connected. This is the notion of connectedness that we wish to investigate.
Definition 18.1.
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.
If a set \(X\) is not connected, we say that \(X\) is disconnected. If \(X\) is a disconnected topological space, then there exist disjoint nonempty open sets \(U\) and \(V\) such that \(X = U \cup V\text{.}\)
Preview Activity 18.1.
Can the subset \(A\) of the topological space \(X\) be written as the union of two disjoint nonempty relatively open sets?
(a)
The set \(A = \{a, b\}\) in \((X, \tau)\) with \(X = \{a, b, c, d\}\) and \(\tau = \{\emptyset, \{a\}, \{b\}, \{a, b\}, X\}\text{.}\)
(b)
The set \(A = \{a, b, c\}\) in \((X, \tau)\) with \(X = \{a,b,c,d,e,f\}\) and
\begin{equation*}
\tau = \{\emptyset, \{a\}, \{c,d\}, \{a, c, d\}, \{b,c,d,e,f\}, X\}\text{.}
\end{equation*}
(c)
The set \(A = X\) with \(X = \{a,b,c,d\}\) and
\begin{equation*}
\tau = \{\emptyset, \{b\}, \{c\}, \{a,b\}, \{b,c\}, \{a,b,c\}, \{b,c,d\}, X\}\text{.}
\end{equation*}
(d)
The set \(A = \{d, f\}\) in \(X = \{a, b, c, d, e, f\}\) with the discrete topology. Generalize your findings.
(e)
The set \(A = \{a, c, d\}\) in \(X = \{a, b, c, d, e\}\) with the indiscrete topology. Generalize your findings.
(f)
The set \(A = \Z\) in \(X = \R\) with the finite complement topology. Generalize your findings.
(g)
The set \(A = X\) in \(X = \{x \in \R \ \mid \ 1 \leq x \leq 2 \text{ or } 3 \lt x \lt 4\}\) with the subspace metric topology from \((\R, d_E)\text{.}\)
(h)
The set \(A = X\) in \(X = \left\{(x, y) \in \R^2 \ \mid \ y = \frac{1}{x} \text{ or } y = 0\right\}\) with open sets
\begin{equation*}
\tau = \{U \cap X \ \mid \ U \text{ is open in the Euclidean Topology on } \R^2\}\text{.}
\end{equation*}
