Definition:Path-Connected

Let $$T$$ be a topological space.

Then $$T$$ is path-connected iff every two points in $$T$$ can be joined by a path in $$T$$.

That is, $$T$$ is path-connected if, for every $$x,y \in X, \exists$$ a continuous mapping $$f:[0,1] \to X$$ such that $$f(0)=x \ $$ and $$f(1)=y \ $$.