Connectedness says a space cannot be split into two separated open pieces. Path connectedness is stronger and uses continuous intervals to join points.
Separations and connected spaces
A separation is a genuine topological tear.
Definition. A separation of \(X\) is a pair of disjoint nonempty open sets \(U,V\) with \(X=U\cup V\). A space is connected if it has no separation.
Proposition. A space \(X\) is connected if and only if the only subsets that are both open and closed are \(\varnothing\) and \(X\).
Proof. If \(A\) is nonempty, proper, open, and closed, then \(A\) and \(X\setminus A\) form a separation. Conversely, if \(U,V\) form a separation, then \(U\) is nonempty, proper, open, and also closed because its complement is \(V\).
Continuous images and intervals
Connectedness is preserved by continuous maps and explains the intermediate value theorem.
Theorem. The continuous image of a connected space is connected.
Proof. Let \(f:X\to Y\) be continuous and suppose \(f(X)=A\cup B\) is separated by disjoint nonempty open sets in the subspace \(f(X)\). Then \(f^{-1}(A)\) and \(f^{-1}(B)\) are disjoint nonempty open sets covering \(X\), contradicting connectedness.
Corollary. If \(X\) is connected and \(f:X\to\mathbb R\) is continuous, then \(f(X)\) is an interval.
Proof. Connected subsets of \(\mathbb R\) are intervals. Indeed, if \(a<c<b\) with \(a,b\in E\subseteq\mathbb R\) and \(c\notin E\), then \(E\cap(-\infty,c)\) and \(E\cap(c,\infty)\) separate \(E\).
Path connectedness
A path is a continuous parametrized bridge from one point to another.
Definition. A path from \(x\) to \(y\) in \(X\) is a continuous map \(\gamma:[0,1]\to X\) with \(\gamma(0)=x\) and \(\gamma(1)=y\). A space is path connected if every two points can be joined by a path.
Proposition. Every path connected space is connected.
Proof. If \(X=U\cup V\) were a separation and \(x\in U,y\in V\), any path \(\gamma\) from \(x\) to \(y\) would have connected image \(\gamma([0,1])\). But this image would meet both \(U\) and \(V\), producing a separation of the image. This contradicts connectedness of \([0,1]\).
Example. The topologist's sine curve closure is connected but not path connected. Thus path connectedness is strictly stronger.
Components
Components decompose a space into maximal connected pieces.
Definition. The connected component of \(x\in X\) is the union of all connected subsets of \(X\) containing \(x\). Path components are defined similarly using path connected subsets.
Proposition. Connected components are connected, closed, and form a partition of \(X\).
Proof. A union of connected subsets with a common point is connected, so the component is connected. Maximality makes two components either equal or disjoint, giving a partition. The closure of a connected set is connected, so the closure of a component is connected and hence contained in the component; therefore the component is closed.
Remark. Components need not be open. They are open in locally connected spaces, a condition often satisfied by manifolds but not by arbitrary spaces.