Let\(Y\) be a subset of a topological space \(X\) . A pair of open sets \(\setList{A,B}\) satisfying \(A\cap Y\not=\emptyset\text{,}\)\(B\cap Y\not=\emptyset\text{,}\)\(Y\subseteq A\cup B\text{,}\) and \(A\cap B\cap Y=\emptyset\) is called a disconnection of \(Y\text{.}\)
A subset of a space for which a disconnection exists is called disconnected; otherwise, the space is called connected.
Definition7.2.
A set which is both closed and open is said to be clopen.
Proposition7.3.
A pair \(\setList{A,B}\) of open sets is a disconnection of \(Y\subseteq X\) if and only if \(\setList{A\cap Y,B\cap Y}\) is a partition of \(Y\) by non-empty clopen sets in the subspace topology.
Corollary7.4.
A space itself is disconnected if and only if it is the union of two disjoint non-empty clopen subsets.
Proposition7.5.
The Euclidean line with a point removed \(\mb R\setminus\setList{0}\) is disconnected.
Lemma7.6.
Let \(\mb R=U\cup V\) for open sets \(U,V\) and let \(x\in U,y\in V\) with \(x\leq y\text{.}\) Then \(\inf\setBuilder{z\in[x,y]}{z\in V}\in U\cap V\text{.}\)
Corollary7.7.
The Euclidean line is connected.
Proposition7.8.
The Sorgenfrey topology on \(\mb R\) is disconnected.
Theorem7.9.
If a subset \(A\) of a topological space is connected, then \(\cl A\) is connected.
Proposition7.10.
If a subset \(A\) of a topological space \(X\) is connected and \(f:X\to Y\) is continuous, then \(f[A]\) is connected.
Corollary7.11.
Connectedness is a topological property.
Proposition7.12.
Let \(\setList{0,1}\) have the discrete topology. Then a topological space \(X\) is connected if and only if every continuous function \(f:X\to\setList{0,1}\) is constant.
Theorem7.13.
If \(\mc A\) is a collection of connected subsets of a topological space with \(\bigcap\mc A\not=\emptyset\text{,}\) then \(\bigcup\mc A\) is connected.
Definition7.14.
Suppose for every two points \(x,y\in A\subseteq X\text{,}\) there exists a continuous function \(f:[0,1]\to A\) such that \(f(0)=x\) and \(f(1)=y\text{.}\) Such a space is said to be path connected.
Proposition7.15.
Every path connected space is connected.
Checkpoint7.16.
Find or create an example of a connected space that's not path connected.
Theorem7.17.
Let
\begin{equation*}
S=
\setBuilder{\tuple{x,y}}{x\in(0,1]\text{ and }
y=\sin\left(\frac{1}{x}\right)}
\end{equation*}
(the topologist's sine curve). Then \(\cl S\) is a subset of the Euclidean space \(\mb R^2\) that is connected but not path connected.
