== Definition == A topological space X {\displaystyle X} is said to be path connected if for any two points x 0 , x 1 ∈ X {\displaystyle x_{0},x_{1}\in X} there exists a continuous function f : [ 0 , 1 ] → X {\displaystyle f:[0,1]\to X} such that f ( 0 ) = x 0 {\displaystyle f(0)=x_{0}} and f ( 1 ) = x 1 {\displaystyle f(1)=x_{1}} == Example == All convex sets in a vector space are connected because one could just use the segment connecting them, which is f ( t ) = t a → + ( 1 − t ) b → {\displaystyle f(t)=t{\vec {a}}+(1-t){\vec {b}}} . The unit square defined by the vertices [ 0 , 0 ] , [ 1 , 0 ] , [ 0 , 1 ] , [ 1 , 1 ] {\displaystyle [0,0],[1,0],[0,1],[1,1]} is path connected. Given two points ( a 0 , b 0 ) , ( a 1 , b 1 ) ∈ [ 0 , 1 ] × [ 0 , 1 ] {\displaystyle (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 + t a 1 , ( 1 − t ) b 0 + t b 1 ] {\displaystyle f(t)=[(1-t)a_{0}+ta_{1},(1-t)b_{0}+tb_{1}]} for t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} .The preceding example works in any convex space (it is in fact almost the definition of a convex space). == Adjoining Paths == Let X {\displaystyle X} be a topological space and let a , b , c ∈ X {\displaystyle a,b,c\in X} . Consider two continuous functions f 1 , f 2 : [ 0 , 1 ] → X {\displaystyle f_{1},f_{2}:[0,1]\to X} such that f 1 ( 0 ) = a {\displaystyle f_{1}(0)=a} , f 1 ( 1 ) = b = f 2 ( 0 ) {\displaystyle f_{1}(1)=b=f_{2}(0)} and f 2 ( 1 ) = c {\displaystyle f_{2}(1)=c} . Then the function defined by f ( x ) = { f 1 ( 2 x ) if x ∈ [ 0 , 1 2 ] f 2 ( 2 x − 1 ) if x ∈ [ 1 2 , 1 ] {\displaystyle 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 {\displaystyle a} to c {\displaystyle c} . Thus, a path from a {\displaystyle a} to b {\displaystyle b} and a path from b {\displaystyle b} to c {\displaystyle c} can be adjoined together to form a path from a {\displaystyle a} to c {\displaystyle c} . == Relation to Connectedness == Each path connected space X {\displaystyle X} is also connected. This can be seen as follows: Assume that X {\displaystyle X} is not connected. Then X {\displaystyle X} is the disjoint union of two open sets A {\displaystyle A} and B {\displaystyle B} . Let a ∈ A {\displaystyle a\in A} and b ∈ B {\displaystyle b\in B} . Then there is a path f {\displaystyle f} from a {\displaystyle a} to b {\displaystyle b} , i.e., f : [ 0 , 1 ] → X {\displaystyle f:[0,1]\rightarrow X} is a continuous function with f ( 0 ) = a {\displaystyle f(0)=a} and f ( 1 ) = b {\displaystyle f(1)=b} . But then f − 1 ( A ) {\displaystyle f^{-1}(A)} and f − 1 ( B ) {\displaystyle f^{-1}(B)} are disjoint open sets in [ 0 , 1 ] {\displaystyle [0,1]} , covering the unit interval. This contradicts the fact that the unit interval is connected. == Exercises == Prove that the set A = { ( x , f ( x ) ) | x ∈ R } ⊂ R 2 {\displaystyle A=\{(x,f(x))|x\in \mathbb {R} \}\subset \mathbb {R} ^{2}} , where f ( x ) = { 0 if x ≤ 0 sin ( 1 x ) if x > 0 {\displaystyle f(x)=\left\{{\begin{array}{ll}0&{\text{if }}x\leq 0\\\sin({\frac {1}{x}})&{\text{if }}x>0\\\end{array}}\right.} is connected but not path connected.