connectednessconvex-analysisgeneral-topologyreal-analysis
What is the difference between a convex set and a simply connected
set, as a subset of topological space, and as a subset of
$\mathbb{R}^n$ in the usual topology ?
I mean, as far as I understood, a convex connected set is simply connected and vice versa, but if this were to be true, why there are two notion with different names ?
Note that the emphasise of the question is on the fact that the set is simply connected. I mean of course a convex set is connected, but what about its simpleness ?
A locally connected space is a space where every neighborhood of every point $x$ contains an connected open subset containing $x$. Informally, something preventing a space from being locally connected would be a neighborhood of a point with "disconnected parts arbitrarily close to that point". Here is an attempt at a drawing of a connected space which is not locally connected (the dashed red line is a neighborhood of the red mark showing that the space is not locally connected). It is the comb space.
Best Answer
Simply connected means something else: that any loops inside the set can be continuously deformed to a point (as well as the set bring connected in the usual sense). So, for instance, an annulus in $\Bbb R^2$ is not simply connected because a loop going around the ring can't be continuously deformed to a point.
A convex set in $\Bbb R^n$ must be simply connected, but a simply connected set needs not be convex. For instance, a star-shaped set in the plane is simply connected but not convex.