# Formal and intuitive meaning of the simple connectedness

A topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question.

I understand both formal and intuitive meaning of the simple connectedness, but I can't get the idea how they imply each other. For example, given the following 2D-picture of a space that is connected but not simply connected. I've tried to draw two paths that contradict to simple connectedness. I can't understand why we can't build a homeomorphism between them. • Can you show us such an homeomorphism ? What are the intermediate curves ?
– user65203
Feb 3 '20 at 18:08
• Homeomorphism is totally the wrong word here. The paths are, of course, homeomorphic, but they are NOT homotopic. Feb 3 '20 at 18:48
• @TedShifrin thanks for the correction! Feb 3 '20 at 19:14

Recall that a (path) homotopy between a path $$\alpha: p_1 \rightsquigarrow p_2$$ and a path $$\beta: p_1 \rightsquigarrow p_2$$ in a topological space $$X$$ is a continuous function $$H: [0,1] \times [0,1] \rightarrow X$$ s.t. $$H(x,0) = \alpha(x)$$ and $$H(x,1) = \beta(x)$$ and also $$H(0,t) = p_1$$ and $$H(1,t) = p_2$$.
One may also interpret this as a family of paths parametrized by $$t \in [0,1]$$ s.t. the starting point and the endpoint are the same for any of those paths, which varies continuously w.r.t. the parameter $$t \in [0,1]$$.
So in particular, the images of any of those paths needs to be contained in $$X$$. If you wanted to draw such a family of curves in your picture, they would either have to be discontinuous, of transition through the hole in the space. Either of which is a contradiction.