Is this a simply connected set and how can we prove?

Question

I get the intuitive idea of a simply connected set ( any closed curve could be shrinked to a point). Can you please state the precise definition of a simply connected set and how can we use it to prove that this set is simply connected ?

As I feel this set goes against the intuitive idea of a simply connected set?

Set is $\mathbb{R^2}$ \ {$(0,0)$} .

Answer 1

If you are in (multivariable) calculus, the definition usually goes like this:

Defintion: A plane region $D$ is simply connected if it is connected and every simple closed curve in $D$ surrounds only points in $D$.

In your case, the unit circle traces out a simple closed curve in $D=\mathbb{R}^2-\{(0,0)\}$. However, that curve surrounds the origin, which is not a point in $D$. Hence $D$ is not simply connected (though it is connected).

Answer 2

First, you can talk about a simply connected topological space, not about a simply connected set.

The definition is that a space $X$ is simply connected if

1. it is connected
2. for every continuous map $f:\Bbb S^1\to X$ there is a map $g:\Bbb D^2\to X$ such that $f=g|_{\partial \Bbb D^2}$.

The second condition can be roughly worded as "every closed curve can be shrunk to a point", like you said. The continuous image of a disk is what tells you how to shrink its boundary to its center.