Topology/Path Connectedness

Definition
A topological space $$X$$ is said to be path connected if for any two points $$x_0, x_1\in X$$ there exists a continuous function $$f:[0,1]\to X$$ such that $$f(0)=x_0$$ and $$f(1)=x_1$$

Example
The preceding example works in any convex space (it is in fact almost the definition of a convex space).
 * 1) All convex sets in a vector space are connected because one could just use the segment connecting them, which is $$f(t)=t\vec{a}+(1-t)\vec{b}$$.
 * 2) The unit square defined by the vertices $$[0,0], [1,0], [0,1], [1,1]$$ is path connected. Given two points $$(a_0, b_0), (a_1,b_1)\in [0,1]\times[0,1]$$ the points are connected by the function $$f(t)=[(1-t)a_0+ta_1,(1-t)b_0+tb_1]$$ for $$t\in[0,1]$$.

Adjoining Paths
Let $$X$$ be a topological space and let $$a,b,c\in X$$. Consider two continuous functions $$f_1,f_2:[0,1]\to X$$ such that $$f_1(0)=a$$, $$f_1(1)=b=f_2(0)$$ and $$f_2(1)=c$$. Then the function defined by

$$f(x) = \left\{ \begin{array}{ll} f_1(2x) & \text{if } x \in [0,\frac{1}{2}]\\ f_2(2x-1) & \text{if } x \in [\frac{1}{2},1]\\ \end{array} \right.$$

Is a continuous path from $$a$$ to $$c$$. Thus, a path from $$a$$ to $$b$$ and a path from $$b$$ to $$c$$ can be adjoined together to form a path from $$a$$ to $$c$$.

Relation to Connectedness
Each path connected space $$X$$ is also connected. This can be seen as follows:

Assume that $$X$$ is not connected. Then $$X$$ is the disjoint union of two open sets $$A$$ and $$B$$. Let $$a\in A$$ and $$b\in B$$. Then there is a path $$f$$ from $$a$$ to $$b$$, i.e., $$f:[0,1]\rightarrow X$$ is a continuous function with $$f(0)=a$$ and $$f(1)=b$$. But then $$f^{-1}(A)$$ and $$f^{-1}(B)$$ are disjoint open sets in $$[0,1]$$, covering the unit interval. This contradicts the fact that the unit interval is connected.

Exercises
is connected but not path connected.
 * 1) Prove that the set $$A=\{(x,f(x))|x\in\mathbb{R}\}\subset\mathbb{R}^2$$, where $$f(x) = \left\{ \begin{array}{ll} 0 & \text{if } x \leq 0\\ \sin(\frac{1}{x}) & \text{if } x > 0\\ \end{array} \right.$$