This question is from the perspective of complex analysis, with basic knowledge of topology. Let $\Omega$ be an open, path connected set in the complex plane ($\Omega$ being called a region). I have come across two definitions of simply connected regions :  A region $\Omega$ is simply connected if any two pair of curves in Ω with the same end-points are homotopic. (This is from Stein-Shakarchi)  A region $\Omega$ is simply connected if every closed curved in Ω is homotopic to a point in D. (This is from Mathematical Analysis by Apostol)   Its evident that definition-1 implies definition-2. But I am not able to see how definition-2 implies definition-1 ? Any help would be appreciated
A topological space $X$ is simply connected if it is pathwise connected and each closed path $u : I = [0,1] \to X$ is path homotopic to the constant path at $x_0 = u(0) = u(1)$. Recall that  A closed path $u$ is one with $u(0) = u(1)$. A closed path will also be called a loop and we write $\mathcal L'(X)$ for the set of all loops in $X$. If we want to be more precise, we say $u$ is a loop based at $x_0 \in X$ provided $x_0 = u(0) = u(1)$. By  $\mathcal L'(X,x_0)$ we denote the set of all loops in $X$ which are based at $x_0$. Also note that each constant path is a loop.  Two paths  $u, v : I \to X$ with the same initial point $x_0 = u(0) = v(0)$ and the same terminal point $x_1 = u(1) = v(1)$ are path homotopic, $u \simeq_p v$, if there exists a  homotopy $H : I \times I \to X$ such that $H(t,0) = u(t)$ and $H(t,1) = v(t)$ for all $t$ and $H(0,s) = x_0, H(1,s) = x_1$ for all $s$ (such a homotopy $H$ is stationary on the boundary $\partial I = \{0,1\}$; it is called a path homotopy). It is well-known that $\simeq_p$  is an equivalence relation on the set $\mathcal P(X)$ of all paths in $X$. Clearly $\simeq_p$ restricts to an equivalence relation on $\mathcal L'(X)$ and to an equivalence relation on each $\mathcal L'(X,x_0)$.  The set of equivalence classes $\mathcal L'(X,x_0)/\simeq_p$ is nothing else than the fundamental group $\pi_1(X,x_0)$ with base point $x_0$.   Question: Find a list, as complete as possible, of properties characterizing topological spaces which are simply connected. Each of these properties can then be taken as an alternative definition of simply connected. Note that many questions in this forum deal with aspects of simple connectedness. The answer to the present question will hopefully cover most of them.
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